TAMPA, Fla. — Prognosticators have issued a grim assessment of Donald Trump’s chances, but the GOP nominee’s path to the White House is still alive and runs through Florida, where his campaign has gained momentum in the final stretch of the race.

He’s rising in the polls in the Sunshine State and remains competitive in Ohio — the two biggest battleground prizes on the map — and the volatility of the 2016 election leaves his backers confident that he holds the advantage thanks to an enthusiasm gap that the tycoon holds over Democratic nominee Hillary Clinton.

“Anybody who says the race is over — that’s preposterous,” said Peter Feaman, a GOP national committeeman in Florida. “Now that early voting has started and the turnout so far is setting records, that bodes well for Trump, not Clinton, because the enthusiasm for Trump is through the roof.”

Mrs. Clinton’s crowds are growing as the Nov. 8 election draws near, including several thousand who showed up for a midday rally here Wednesday. But she didn’t approach the more than 20,000 people who two days earlier filled the nearby MidFlorida Credit Union Amphitheater to see Mr. Trump.

“It’s unbelievable. I’ve never seen anything like this,” Mr. Feaman said of the rock concert-size crowds. “When Mitt Romney ran in 2012, we would struggle to get 500 people in an abandoned airplane hangar to make it look crowded.”

The map is still a major challenge for Mr. Trump. Even if he prevails in Florida and Ohio, he still must cobble together wins in states such as Iowa, Nevada, New Hampshire, North Carolina and Wisconsin, as well as hold on to all of the states won by Mr. Romney in 2012.

The latest polling shows the GOP nominee struggling in Arizona, Georgia and Utah, all states that George W. Bush and Mitt Romney won with ease.

Indeed, GOP vice presidential candidate Mike Pence campaigned in Utah on Wednesday while Mr. Trump was in Washington, D.C. — two places where a successful Republican ticket shouldn’t be spending time with less than two weeks to go before Election Day.

Mr. Pence was urging conservative Utah voters to stick with the GOP, while Mr. Trump was tending to his day job running his business empire. He attended the grand opening of his new luxury hotel, which was refurbished from the historic Old Post Office just blocks from the White House.

The ribbon-cutting ceremony, however, fueled questions about whether he was wasting some of the precious little time he has left before Election Day. Mr. Trump said he was trying to show the country what he can do, adding that the hotel came in under budget and ahead of schedule.

“We turned a property that had been neglected for decades, and which was losing huge sums of money for the federal government, into a major revenue producer and job creator,” said the billionaire businessman, standing before a lectern emblazoned with the “Trump Hotels” logo. “This is what I want to do for our country. And this is what we’re working so hard to do.”

In television interviews and a campaign stop later in the day in North Carolina, Mr. Trump projected confidence.

“New polls are coming out. We’re leading Florida. We’re doing great in North Carolina. We’re doing great in Pennsylvania. We’re doing great all over,” Mr. Trump told CNN. “We’re doing really well in New Hampshire — Ohio, as you know, and Iowa are doing fantastically well. I’m telling you, CNN doesn’t say it, but I think we’re going to win.”

The GOP ticket got a boost Wednesday from a Bloomberg Politics poll that showed him leading Mrs. Clinton by 2 points in Florida, 45 percent to 43 percent.

The pollsters credit Florida independent voters with shifting to Mr. Trump to put him in front.

It was the first Florida poll he has topped in more than three weeks, and the Real Clear Politics rolling average of recent polls in Florida, which includes the Bloomberg Politics poll, still puts Mrs. Clinton ahead of Mr. Trump by 1.6 points, 46.4 percent to 44.8 percent.

Florida is a must-win for Mr. Trump because he would have virtually no path to the White House without the state’s 29 electoral votes. Mrs. Clinton has alternative paths to the 270 electoral votes needed to win, but she has campaigned hard in Florida in an effort to block Mr. Trump and sew up a victory.

Both candidates barnstormed across the state this week as early voting got underway.

Mrs. Clinton, warning that Mr. Trump poses an existential threat to American values, urged her supporters to get to the voting locations as soon as possible.

“Donald Trump says he can still win, and he’s right,” she told the huge crowd gathered in a downtown park beside the Hillsborough River. “That’s why it’s so important you get out there and vote.”

Mr. Trump delivered a similar appeal to the throngs at the amphitheater, where a large share of the crowd was women and Hispanics, two groups that Mrs. Clinton hopes to dominate at the ballot box.

In the crucial battleground state of Ohio, Mr. Trump has consistently led or tied in recent polls. The Real Clear Politics average of Ohio polls showed him ahead of Mrs. Clinton by 1.1 points, 44.8 percent to 43.7 percent.

But Mr. Trump’s early-campaign predictions of expanding the electoral map to New York and California haven’t come to pass. Instead, his path to the White House has narrowed in recent weeks, according to political handicappers.

The Rothenberg & Gonzales Political Report, a nonpartisan election tracker, showed Mrs. Clinton positioned to win 323 electoral votes versus 191 for Mr. Trump, with 24 tossup votes on the line in Iowa and Ohio.

The Cook Political Report gave Mrs. Clinton a 278-to-179 vote edge, with 81 tossup votes, and the University of Virginia’s Center for Politics gives Mrs. Clinton a 353-to-173 lead with 13 more up for grabs in Iowa and Utah.

⦁ Seth McLaughlin reported from Washington.

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