Cummins, 50, and the teen threw their phones in a river and disabled their car's GPS device to avoid detection, FBI agent Utley Noble said.
The two went missing on March 13, and were found on April 20 at a remote cabin in Northern California, ending a frenzied search that involved law enforcement agencies across the country.
Cummins is charged with
taking a minor across state lines for the purpose of sexual activity. He has not yet entered a plea. If convicted, he faces 10 years to life in prison.
Cummins will remain in jail until his trial, a federal judge ruled Friday. Assistant US Attorney Sara Beth Myers presented Noble as a witness at a detention hearing.
Plans to paddle to Mexico
Cummins told authorities he had sexual contact with the teen "most nights" during their time on the run, Noble testified.
While on the run, he bought an electronic tablet to keep up with news of their disappearance, along with a two-person kayak worth $1,500, Noble said.
The pair planned to paddle to Mexico from Coronado Island in San Diego, he said. They used aliases of a married couple, John and Joanne Castro, so they would blend in once they crossed the border.
Before ending up in California, they made stops in Alabama, where they threw their phones into the Tennessee River, federal authorities said. They also allegedly switched license plate tags from Tennessee to Alabama, then from Alabama to Colorado.
Before he disappeared, Cummins left his wife a note saying he had gone to Virginia Beach to clear his head, the authorities said.
"Don't call the police. They'll think I ran because I'm guilty and I'm not!" reads the note, which was entered into the court record
'I can't explain what he's done'
Friday's hearing was held to address Cummins' request to be released from custody while he awaits trial.
The request was denied by Magistrate Barbara Holmes, who deemed Cummins a flight risk and a danger to the public. The case is being heard
in the Middle District of Tennessee.
Cummins' sister, Daffney Quinn, attended the hearing, along with his father, cousins and two daughters.
Quinn defended Cummins' character and tearfully described him as "the best brother a sister could ask for."
"I can't explain what he's done," she testified. "Something has happened psychologically and mentally to cause what he's done."
"I strongly believe that something is wrong with him, because the brother and man I know would never do something like this."
If Cummins were released before his trial, Quinn said his family would supervise him constantly and would be the first to turn him in if he broke any probation rules.
Asked if her brother has expressed regret, she paused.
"No. Not necessarily," she said. "I don't think he knew this would happen or realized he would be in this much trouble."
After the hearing, Cummins' cousins, Tony and Jerry Brawner, said the most important thing is for the girl to get help.
"It's a horrible thing he's done, and we don't condone his actions," they said. "But he's family, and we still love him."
Cummins and the girl disappeared March 13 as a police investigation into their relationship was heating up.
The former high school health sciences teacher in Culleoka, Tennessee, had been suspended in February, less than a month after a student reported seeing him and the teenage girl kissing in a classroom.
Authorities believe the teacher groomed the girl for nearly a year before they vanished.
According to a criminal complaint
, Cummins refilled a prescription for Cialis, which is used to treat erectile dysfunction; took out a cash loan; and made hotel reservations in Oklahoma. Investigators argue those actions show he was planning to leave with the girl.
Cummins was fired the day after they disappeared.
Cummins also faces state charges of sexual contact with a minor and aggravated kidnapping, the Tennessee Bureau of Investigation said last month.
He has pleaded not guilty.
Tennessee law allows children older than 12 to decide whether to leave their families unless their removal or confinement "is accomplished by force, threat or fraud."
To prove kidnapping, prosecutors would need to show the teen was unlawfully removed or had her freedom restricted.