Governor DeSantis and a former ambassador to the United Nations, Nikki Haley, will face off in two debates this month ahead of the Iowa caucuses and the New Hampshire primary. Mr. DeSantis wants voters to help him defy the odds in Iowa, and Ms. Haley is leaning on independent voters to try to secure a win in New Hampshire.

On the national level, the GOP primary appears to be all but locked up for President Trump, who leads the second-place candidate, Mr. DeSantis, by about 50 points, according to the FiveThirtyEight average of national polls.

In Iowa and New Hampshire, though, Mr. DeSantis and Ms. Haley are working overtime to try to pull off an unlikely win in the hopes of building momentum in the national competition.

The candidates will have what may be their last best shot to get their message out to voters in Iowa and New Hampshire later this month, with debates hosted by CNN in Iowa and New Hampshire on January 10 and 21, respectively.

In Iowa, Mr. DeSantis has been on the ground in a blitz between Christmas and New Year’s, asking a crowd at an event Sunday evening, New Year’s Eve, “Are you ready to work hard over these next two weeks and win the Iowa caucuses?” 

Mr. Trump and his campaign have also moved to barnstorm the Hawkeye State, with 12 events scheduled there ahead of the January 15 caucuses.

Before Christmas, Mr. Trump was encouraging his supporters to not take victory in the state for granted, saying at an event at Waterloo, “The key is, you have to get out and vote.”

Recent polling in Iowa suggests that Mr. Trump enjoys about a 30-point lead in the state, with an Insider Advantage poll from before the holidays indicating that he leads Mr. DeSantis by 49 percent to 17 percent.

As Mr. DeSantis goes all in on Iowa, Ms. Haley has adopted a similar strategy in New Hampshire, where she is holding events in the run-up to the primaries on January 21 and has been touting an endorsement from the state’s Republican governor, Chris Sununu.

“A president should bring out the best in all of us — that’s Nikki Haley,” Mr. Sununu said in an ad for Ms. Haley at New Hampshire. “She’s a new generation of conservative leadership who can help leave behind the chaos and the drama of the past.”

Mr. Trump has leveled a volley of attacks against Ms. Haley, dubbing her “high tax Haley” in a new ad airing at New Hampshire and buying $3.4 million of air time for the ad in the Granite State, according to Federal Elections Commission filings.

In New Hampshire, most polls show Mr. Trump far ahead of the rest of the field. FiveThirtyEight’s average has Mr. Trump leading Ms. Haley by 44 percent to 26 percent.

Some polling, though, suggests that Ms. Haley could outperform expectations if she is able to convince undeclared voters or Democratic voters at New Hampshire to cast a ballot in the state’s primary.

A recent Saint Anselm College survey found that, though Ms. Haley trails Mr. Trump 24 percent to 56 percent among voters who identify as Republicans and intend to vote in the GOP primary, she leads him 53 percent to 17 percent among those who self-identify as swing voters and 43 percent to 3 percent among those who self-identify as Democratic voters and who intend to vote in the GOP primary.

Although the window has closed for registered Democrats to change their party affiliation to vote in the GOP primary, the New Hampshire secretary of state announced last year that about 4,000 voters had changed their party affiliation ahead of the deadline to vote in the GOP primary, an indication that those former Democrats plan to vote in the GOP primary.

More significant than Democrats changing party affiliation to vote in the GOP primary at New Hampshire will be turnout among unaffiliated voters. Unaffiliated voters are the largest voting block at New Hampshire, according to the Independent Voter Project, with about 399,000 voters there not belonging to any political party. For comparison, there are just 298,000 Republican voters at New Hampshire.

The managing editor of Sabato’s Crystal Ball at the University of Virginia’s Center for Politics, Kyle Kondik, tells the Sun that with a “very sleepy Democratic primary that doesn’t even feature Joe Biden on the ballot, it’s easy to imagine voters who don’t plan to vote Republican for president in November nonetheless crossing over and voting in the GOP presidential primary.”

“Of all the early contests, Trump seems the most vulnerable in New Hampshire, both because of this dynamic and also because the New Hampshire electorate is not that religious and not that conservative compared to the broader Republican primary electorate nationally,” Mr. Kondik says.

Mr. Kondik cautions, though, that even an outright win for Ms. Haley in New Hampshire might not give her campaign the momentum it would need to carry any other early states.

“If Trump were to lose there, it could be akin to George W. Bush losing to John McCain in New Hampshire in 2000. But that also means that even if Trump were to lose New Hampshire, it might not have much impact on his ability to win the nomination,” Mr. Kondik says. “That was the case for Bush in 2000, who easily won the nomination despite the New Hampshire setback.”

The president of the polling firm American Research Group Inc., Dick Bennet, tells the Sun he “could see Nikki Haley beating Trump here” because voters have the ability “to pick and choose the primary.”

A survey from his firm conducted ahead of Christmas found Ms. Haley trailing Mr. Trump in the state 29 percent to 33 percent, a significantly closer result than other pollsters have found and within the poll’s plus or minus 4 point margin of error.

The survey, which was conducted by a random sample of phone interviews, found that 40 percent of respondents who planned to vote in the GOP primary in New Hampshire were undeclared voters. He characterized many of these voters as being fiscally conservative and socially moderate former Republicans.

“I think there’s strategic voting that’s going to go on,” Mr. Bennet. “They tell us they’re done with the Trump stuff, we’ll see.”

A spokeswoman for Ms. Haley’s campaign, AnnMarie Graham-Barnes, tells the Sun that the campaign is “certainly not underestimating the power of the independents,” adding, “We have an effective grassroots campaign doing the door knocking.”

What Governor Christie chooses to do ahead of the GOP primary in New Hampshire could also have an impact on Ms. Haley’s chances of an upset victory.

According to the Saint Anselm College poll, Mr. Christie is the preferred candidate of 12 percent of respondents. Among respondents who have a negative opinion of Mr. Trump, 31 percent support Mr. Christie and 56 percent choose Ms. Haley.

Although Mr. Christie’s supporters are more likely to support President Biden in a general election contest between Messrs. Biden and Trump compared to Ms. Haley’s supporters, a decision to drop out of the race from Mr. Christie could help Ms. Haley carve a path to victory in New Hampshire.