Florida’s Republican Gov. Rick Scott declared victory in his Senate bid for a second time Thursday after a state-mandated machine recount widened his lead over incumbent Democrat Bill Nelson.

But Mr. Nelson’s legal team, insisting there is not “one universe of votes,” vowed Thursday evening to press on and predicted their candidate eventually will emerge as the winner.

While the election results are not official until certified by the secretary of state, a move that should come after the Elections Canvassing Commission’s Nov. 20 meeting, the unofficial recount expanded Mr. Scott’s edge by 865 votes, leaving him with a margin of 13,427.

That’s a bigger edge than any recount in U.S. history has overcome, but it’s less than the 0.25 percent threshold required for a hand recount.

Secretary of State Ken Drezner on Thursday ordered that manual recount for the Senate and Florida’s agricultural commissioner races. It will involve hand reviewing all questionable, provisional or mailed-in ballots that, for one reason or another, had been rejected at the polling place or proved unreadable in the machine.

The governor’s race did not fall within the threshold, so it appears Republican Ron DeSantis will be certified as governor Nov. 20 failing a legal challenge.

Outside the recount, potentially more votes still have to be tallied. For example, while ballots from overseas military and civilians must be postmarked by Nov. 6, the deadline for receiving them is Nov. 16. In addition, a flurry of court activity set in motion since Election Day has left some ballots in limbo.

“We never thought there was a silver bullet, one thing,” said Marc Elias, Mr. Nelson’s lead recount lawyer in a conference call with reporters. “There is not one universe where all these votes are coming from.”

In particular, Mr. Elias pointed to the coming hand recount, which he said “very well may reverse it entirely.”

But the Democrats also are seeking to change Florida’s law, which they say contains an absurdity in that the extension granted to overseas voters does not apply to domestic voters who sent in ballots postmarked by Nov. 6.

Earlier Thursday, U.S. District Judge Mark Walker extended the state deadline under which voters can clear up questions about their mailed-in or provisional ballots, usually raised by issues with signatures. The “cures” for these ballots, to use Florida’s term, is now 5 p.m. Saturday, under the terms of Judge Walker’s ruling.

Mr. Scott appealed the ruling, but it was upheld by the 11th Circuit U.S. Court of Appeals just after 5 p.m.

The issue of voters’ signatures caused Mr. Nelson’s campaign to sound like a boardwalk hawker in a fundraising blast Thursday afternoon.

“Has your signature changed over the years since you first registered to vote?” the fundraising email read. “Have a disability that affects your handwriting? Suffer from eyesight loss? Changed your last name? If you answered ‘yes’ to any of the questions above, then you could be denied your right to vote and that’s why our legal challenge of this disenfranchisement law is so important.”

Mr. Elias defended the poring over signatures and the divining of voter intent that will take center stage over the next several days as a crucial element of ensuring “every vote is counted.”

Judge Walker did brush aside a broader request from Mr. Nelson’s lawyers to extend Florida’s 3 p.m. deadline on reporting the machine recounts, calling the state “a laughingstock” in remarks from the bench.

The Democratic supervisor of elections in Palm Beach County, Susan Bucher, had said machine malfunctions were making it difficult for her to hit the deadline, but she said she did make the deadline for the Senate race.

Like her counterpart in neighboring Broward County, Democratic election supervisor Brenda Snipes, Ms. Bucher’s operation has come under heavy criticism for its inability to match the performance of Florida’s other 65 counties, all of which reported no difficulty recording their votes even in those areas ravaged by Hurricane Michael in October.

“We gave it everything we had,” Ms. Bucher told reporters right after the deadline. “I’m very proud of my team. We reported real results, we counted every ballot.”

While Mr. Nelson enjoyed a big lead in heavily Democratic Broward County, he lost more votes in the machine recount than Mr. Scott did, according to county officials.

Nevertheless, it is in Broward County’s rich Democratic fields that Mr. Nelson’s team believes it will hit pay dirt. The county has nearly 2,000 “undervotes” that need to be reviewed, and between Broward and other counties there are “tens of thousands or higher” ballots that should be carefully scrutinized, Mr. Elias said.

The Democrats have amassed an army of 30,000 volunteers to police the manual recount process, while the Scott campaign said it would field more than 17,000.

Mr. Nelson, who is seeking a fourth term and has been largely invisible since Election Day, has insisted he only wants to ensure all votes are counted.

Republicans have accused Mr. Nelson and his lawyers of trying to steal the election and playing a long game to lay the legal groundwork for any challenges to the 2020 presidential election.

“Everyone knows how this is going to end,” Scott for Florida spokesman Chris Hartline said Thursday afternoon. “Everyone’s in on the joke here. What the Democrats want to do here is throw out all the fraud protections on the books, not re-elect Bill Nelson.”

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