(CNN) — Last week's winter weather travel mess is lingering like a vicious hangover into this week -- and the headaches have been migraine-proportioned for Southwest Airlines, its CEO Bob Jordan, airline employees and most of all its frustrated passengers on Monday.
A total of 4,002 flights within, into or out of the US were canceled on Monday, according to flight tracking website FlightAware, while 9,259 flights were delayed.
But Southwest accounts for a whopping share of those. None of the other US carriers have canceled nearly as many flights or as much of their schedule as Southwest.
The Dallas-based airline had canceled 71% of its flights -- j2,909 total -- on Monday, according to FlightAware. At one point, the airline canceled around 300 flights in the span of a half hour Monday afternoon.
And it looks like the Southwest pain will spread into Tuesday.
More than 2,400 of those flights were already canceled as of 10:50 p.m. ET Monday. That's the lion's share of the more than 2,600 flight cancellations reported so far for Tuesday for all US airlines.
CEO: 'A lot of issues in the operation'
On Christmas night, ahead of Monday's meltdown, Jordan told employees the airline has "a lot of issues in the operation right now."
CNN was provided a transcript of the message to Southwest employees by an aviation source.
Jordan told employees, "Part of what we're suffering is a lack of tools. We've talked an awful lot about modernizing the operation, and the need to do that."
Jordan told The Wall Street Journal the company plans to operate just over a third of its schedule in upcoming days to give itself the ability for crews to get into the right positions.
"We had a tough day today. In all likelihood we'll have another tough day tomorrow as we work our way out of this."
Passengers are mad
On social media, customers are complaining loudly about long lines to speak with representatives, problems with lost bags and excessive wait times or busy signals on the airline's customer service telephone lines. One passenger told CNN her family was on the phone for 10 hours with Southwest.
Passenger Ella Silverstein looks upward while waiting in the line to book a flight Monday in front of the Southwest Airlines ticket counter at Denver International Airport.
Hyoung Chang/Denver Post/Getty Images
On Monday evening, Southwest representatives at Denver International Airport made an announcement over airport terminal speakers indicating flight rebooking for stranded passengers would not be available until Friday, according to a CNN team at the airport.
The announcement also said new bookings would not be possible until January 3.
People in line appeared to shrug off the announcement, CNN's Leslie Perrot observed.
She said passengers have told them throughout the day that they're standing in line to speak with a person, as most people have been unsuccessful reaching representatives over the phone and the website is not operational.
Customers wait to rebook their Southwest Airlines flight at Hartsfield-Jackson Atlanta International Airport on Monday.
CNN's Carlos Suarez spoke with frustrated passengers in line at the Southwest ticket counter at Hartsfield-Jackson Atlanta International Airport on Monday afternoon.
He's reported that about 150 customers were waiting in a long line at one point to rebook, with the line snaking around the back of the ticket counter.
'Flight crews and airplanes ... out of place'
A spokesperson for Southwest blamed the recent winter storm for the cancellations.
"As the storm continued to sweep across the country, it continued to impact many of our larger stations and so the cancellations just compiled one after another to 100 to 150 to 1,000," Jay McVay said at a news conference at Houston's William P. Hobby Airport on Monday night.
"With those cancellations and as a result, we end up with flight crews and airplanes that are out of place and not in the cities that they need to be in to continue to run our operations."
"We will do everything that we need to do to right the challenges that we've had right now," he said, including "hotels, ride assistance, vans ... rental cars to try and make sure these folks get home as quickly as possible."
He promised that all customers, even those who had already left the airport or made alternate arrangements on their own, would also be taken care of.
"If you've already left, take care of yourself, do what you need to do for your family, keep your receipts," McVay relayed. "We will make sure they are taken care of, that is not a question."
An announcement made in the airport terminal before the news conference apologized to customers, and said the next available Southwest seats are on Saturday.
The agent said Southwest would be providing buses to area hotels and assured that "we will have sufficient rooms for all customers who are affected by this disruption."
Emphasis on safety
Southwest earlier responded in an emailed statement Monday afternoon emphasizing safety:
"With consecutive days of extreme winter weather across our network behind us, continuing challenges are impacting our Customers and Employees in a significant way that is unacceptable," the statement read.
"We're working with Safety at the forefront to urgently address wide-scale disruption. ... On the other side of this, we'll work to make things right for those we've let down, including our Employees."
Some of the airports seeing the biggest issues have been Denver, Chicago Midway, Baltimore/Washington, Dallas Love Field, Nashville and Las Vegas.
Customers encountered long lines at Southwest counters on Monday at Raleigh-Durham International Airport.
Calls made Monday afternoon by CNN to Southwest's customer service did not go through, so customers couldn't even get in the queue to speak to a representative. However, Southwest told CNN it is "fully staffed to answer calls."
The airline also says, "those whose flights have been canceled may request a full refund or receive a flight credit, which does not expire."
A tweet from Southwest directing customers to self-service options had more than 1,000 replies -- many of them angry -- around 6 p.m. ET.
One of the replies in part read: "Stop blaming the WEATHER! Had to buy a first class ticket on another airline but it TOOK OFF ON TIME! You still have our luggage with medication inside! Can't get through on the phone!"
Ripple effects at Southwest
In the Sunday night message to employees, Jordan said, "We need to be able to produce solutions faster." He said the airline is "committed to and invested in" improving its systems.
The US Department of Transportation issued a statement on Monday's massive flight cancellations by Southwest, saying the agency is "concerned."
"USDOT is concerned by Southwest's unacceptable rate of cancellations and delays & reports of lack of prompt customer service. The Department will examine whether cancellations were controllable and if Southwest is complying with its customer service plan," the agency tweeted.
The president of the union that represents Southwest's flight attendants told CNN's Pamela Brown in an on-air interview Monday that those systems have left its members stranded.
"The phone system the company uses is just not working. They're just not manned with enough manpower in order to give the scheduling changes to flight attendants, and that's created a ripple effect that is creating chaos throughout the nation," Lyn Montgomery, president of TWU Local 556, told CNN.
She said some flight attendants needed to sleep in airports as a result.
Flight crew treatment called 'despicable'
Montgomery expressed outrage on behalf of workers and customers late Monday afternoon in a news release.
"The way Southwest Airlines has treated its flight crews can only be termed 'despicable,'" said Montgomery. "... Believe me, we know about stepping up and putting in long work hours when we are called to do so; we are flight attendants.
"But at this point, the many years of failure by management, despite many unions' demands to modernize, has left flight attendants fatigued, stranded, hungry and cold -- on Christmas!"
"The company's failures are happening year-round, but are particularly egregious on Christmas," Montgomery said. "Our customers struggled with it just as our thousands of flight attendants did."
Kathleen Bangs, a FlightAware spokesperson, said in on-air interview that Southwest's schedule includes shorter flights with tighter turnaround times that are causing some of the problems. "Those turnaround times bog things down," Bangs said.
CNN has reached out to the airline for comment on Jordan's video message.
What can stranded passengers do?
If you've been left in the lurch and your efforts to reach a customer service agent are going nowhere, the founder of Scott's Cheap Flights suggests trying an international number.
"The main hotline for US airlines will be clogged with other passengers getting rebooked. To get through to an agent quickly, call any one of the airline's dozens of international offices" Scott Keyes said.
"Agents can handle your reservation just like US-based ones can, but there's virtually no wait to get through."
A 10-hour phone call
Michelle Perkins, her husband JJ and their six children had their flights canceled while waiting at the airport in Las Vegas on Christmas Eve. The family was headed to Orlando for a surprise holiday trip for the kids, she said.
"Went from hero to zero" after the trip was canceled, said Perkins, who posted on Instagram along with a video of the family preparing for the trip.
Perkins was traveling with kids from ages 17 years old to 15 months. She told CNN that when their flight was canceled, they headed to baggage claim to get their luggage along with their car seats.
"We went to baggage to retrieve our luggage and car seats, waited over an hour with six kids in tears only to be told that ... bags and car seats must continue on to their destination in Orlando."
Perkins said her husband was on a phone call with Southwest Airlines for nearly 10 hours to get their flights refunded and file a claim on their baggage. She added that they were told their luggage was at the Las Vegas airport.
"We went back to the airport to pick it up, and they denied that our luggage was still there, we told them that it was tracked at Las Vegas, so then they changed their story and said it was in a secure location on the tarmac and they cannot get it because there are too many bags."
As of Monday afternoon, they had yet to be reunited with their checked luggage.
Perkins said they are just thankful they were able to go back home to Las Vegas instead of being stranded in another city.
Any relief in sight?
It could be next week before this is all completely sorted out.
"When there's more than 10,000 flight cancellations over the past week, it takes time for airlines to work through and reaccommodate the backlog of travelers," Keyes told CNN Travel in an email.
"While it will depend on the weather forecast (which looks promising for much of the country) and what number of travelers wind up canceling their holiday plans, I'd expect that by next week, things will have largely returned to normal," Keyes said.
And why are so many people having trouble rebooking?
"One complicating factor for people hoping to get reaccommodated is the fact that there are so few available seats this season," Keyes said.
"That's both because Christmas and New Year is one of the most popular times of year for travel, and because the number of flights on the schedule this year is still down 15-20%, making the challenge even steeper for those needing to get rebooked."
Buffalo still reeling
Meanwhile, in hard-hit western New York, Buffalo International Airport said in its most recent tweet that it plans to resume passenger flights at 11 a.m. ET Wednesday, pushing back the expected reopening by another 24 hours than previously anticipated.
The temperature at the airport was 19 degrees Fahrenheit (minus 7 Celsius) around 8 p.m. ET Monday, with light snow falling on top of the huge amounts the area has already seen.
And it's not just the airport affected there. Road travel remained treacherous because of extreme wintry conditions.
In New York state's western Erie County, emergency restrictions on driving were lifted in some communities but remained in place in Buffalo, County Executive Mark Poloncarz said Monday.
"The City of Buffalo is impassable in most areas, while mains may have a lane open for emergency traffic or two, most secondaries as well as side streets have not been touched yet," Poloncarz said.
He adding that the cleared main roads are primarily for the use of life-saving measures to open up areas around hospitals and nursing homes.
A rough past week
A winter storm that swept across the US was ill-timed for travelers who had started pushing Christmas week flying numbers back toward pre-pandemic levels.
On Christmas Day, there were 3,178 flights canceled and 6,870 flight delayed, according to FlightAware.
On Christmas Eve, there were a total of 3,487 flights canceled, according to FlightAware.
Friday was the worst day of this streak with 5,934 cancellations, while Thursday saw almost 2,700 cancellations.
This megablast of winter weather across the eastern two-thirds of the nation is forecast to slowly moderate this week.
CNN's Ross Levitt, Leslie Perrot, Andi Babineau, Amanda Jackson, Rebekah Reiss, Chris Boyette and Artemis Moshtaghian contributed to this story.