Former Gov. Nikki Haley of South Carolina on Tuesday defied calls to drop out of the race for the Republican nomination, vowing to fight on after a second straight defeat at the hands of former President Donald J. Trump.

In rousing remarks, Ms. Haley looked ahead to the coming primary contest in South Carolina, where she is lagging far behind Mr. Trump in polls despite a home-state advantage.

“New Hampshire is first in the nation. It is not the last in the nation. This race is far from over,” Ms. Haley said, adding, “We’re going home to South Carolina.”

Borrowing signature lines from her stump speeches, Ms. Haley noted how far she had come since the race first opened, when she was polling at just over 2 percent, declaring herself “a fighter.”

“And I’m scrappy. And now we’re the last ones standing next to Donald Trump,” she added.

Ms. Haley also turned up the heat on Mr. Trump, the dominant front-runner in the Republican race who is fighting 91 felony charges, criticizing him as equally bad for the country as another four years of President Biden. She also took another dig at Mr. Trump’s mental fitness and his 77 years of age.

“With Donald Trump you have one bout of chaos after another,” she said. “This court case, that controversy, this tweet, that senior moment. You can’t fix Joe Biden’s chaos with Republican chaos.”

In her final Granite State appearances before polls closed, Ms. Haley had rejected claims that Republican voters had already solidly united behind the former president, and pledged not to end her bid no matter the result.

“I didn’t get here because of luck,” she said at a polling site in Hampton, N.H., while flanked by supporters, including Gov. Chris Sununu, her top surrogate in the state. “I got here because I outworked and outsmarted all the rest of those fellas. So I’m running against Donald Trump, and I’m not going to talk about an obituary.”

Mr. Trump, speaking to supporters at his victory party, mocked Ms. Haley for speaking “like she won.” But “she didn’t win — she lost,” he added.

On Wednesday morning, Ms. Haley is expected to speak during a Republican State Committee meeting in the Virgin Islands, which holds its contest on Feb. 8. She is then anticipated at a homecoming rally in Charleston, S.C.

A number of people close to Ms. Haley are encouraging her to keep going, many who are deeply opposed to Mr. Trump’s becoming the nominee again.

Betsy Ankney, her campaign manager, released a memo early Tuesday morning shooting down suggestions that Mr. Trump’s path to the nomination was inevitable. She pointed to the 11 of the 16 states that vote on Super Tuesday that have “open or semi-open primaries” that can include independent voters and are “fertile ground for Nikki.”

Nevada will host a Republican caucus on Feb. 8, but Ms. Haley is not competing in that contest, instead participating in a Republican primary in the state two days earlier that awards no delegates.

Her campaign has bought over $1 million in television advertising from Tuesday through Feb. 6, according to AdImpact, a media-tracking firm.

And officials at her allied super PAC, Stand for America, said they, too, planned to forge ahead.

Mark Harris, the lead strategist for the PAC, said it was prepping television, mail and digital advertising in a get-out-the-vote effort that would look similar to the programs it took on in Iowa and New Hampshire, though as of Tuesday it had not yet made those investments.

“We’re running the outsider candidacy, so this was never going to happen all magically in one day, and so we’re going to keep pushing ahead,” Mr. Harris said.

Since the summer, Ms. Haley has predicted that the Republican nominating contest would result in a showdown between herself and Mr. Trump in her home state. Her outward confidence in that scenario has not faltered — not after she failed to place second in Iowa, not after her top rival for No. 2, Gov. Ron DeSantis of Florida, dropped out and endorsed Mr. Trump, not after a slate of South Carolina legislators this week joined Mr. Trump on the stump in the final days of the New Hampshire race.

Her message to his allies and the news media: She has been here before.

“I won South Carolina twice as governor,” she told reporters Friday at a retro diner in Amherst. “I think I know what favorable territory is in South Carolina.”

Maggie Haberman and Kellen Browning contributed reporting.