(CNN) -- A handful of key counties in the presidential battleground of Florida extended early voting on Sunday due mainly to heavy turnout.
President Barack Obama and Republican challenger Mitt Romney are running neck and neck in polls in the race to capture Florida's 29 electoral votes on Tuesday.
Polling places in Miami-Dade, Palm Beach, Hillsborough, and Pinellas counties were operating under extended hours to handle absentee balloting, the state's Democratic Party said in a statement.
Those areas acted to keep the doors open even though early voting technically ended statewide on Saturday.
Separately, a judge ordered a continuation of early voting at a Winter Park polling station in Orange County after a suspicious package was found there and disrupted the process on Saturday.
"We encourage registered voters in these counties to cast their ballots today," Scott Arceneaux, executive director of the Florida Democratic Party, said in a statement.
He also said it was "still our hope" that a judge would approve the Democrats' request early on Sunday for an injunction to force the state to keep polls going ahead of Election Day in Miami-Dade, Palm Beach and Broward counties even though two of them had already done so.
Their complaint filed in U.S. District Court in Miami said waits of up to seven hours in "many cases" over the weekend were the "result of polling facilities that are inadequate" to meet the needs of early voting. In some cases, lines "deferred or prevented" ballots from being cast, the complaint said.
Republican Gov. Rick Scott has refused earlier requests to extend early voting.
An Obama campaign official previously said that early voting in Leon County had been extended. Democrats later said that was not the case.