Nikki Haley might have come in third in the Iowa caucuses, but as she campaigns in New Hampshire for its first-in-the-nation primary next week, her attention is squarely focused on only one rival: Donald J. Trump.

Ms. Haley, a former South Carolina governor who served as U.S. ambassador to the United Nations under Mr. Trump, has begun fine-tuning her argument against her former boss, trying out new jabs and unleashing a new attack ad right out of the gate. She has also stepped up her efforts to frame herself as Mr. Trump’s top rival, announcing that she would no longer participate in primary debates that don’t include him.

In recent remarks and in a new television ad, Ms. Haley paints Mr. Trump and President Biden as two sides of the same coin: politicians past their prime who are unable to put forth a vision for the country’s future because they are “consumed by the past, by investigations, by grievances.”

At a campaign rally on Wednesday in Rochester, N.H., she fended off Mr. Trump’s attacks on her immigration record, warned voters not to believe his ads against her and reminded them that it was Mr. Trump who had wanted to raise the age for Social Security eligibility and had once proposed increasing the gas tax.

“Those are things he needs to answer for. Oh, that’s right! He won’t get on the debate stage,” she swiped.

In Bretton Woods, N.H., where she held her first rally after the Iowa caucuses on Tuesday, Ms. Haley said she had voted for Mr. Trump twice and was “proud to serve in his administration” before delving into her familiar critique that, whether fairly or not, “chaos follows” the former president. But she also took a few more acidic digs.

“The majority of Americans think that having two 80-year-olds running for president is not what they want,” she said to a smattering of applause at the Omni Mount Washington Resort.

To be sure, Ms. Haley has not entirely abandoned her measured approach to Mr. Trump. On Thursday, she told voters she would not attack him personally — “people either want me to hate Trump or love Trump” — saying people are tired of that kind of politics. “I just tell it on policy,” she said.

She has not flatly rejected running as his vice president and does not talk about his criminal charges. In a CNN interview, Ms. Haley claimed she had not “paid attention to his cases” when the host, Dana Bash, asked how she felt about her party’s front-runner being held liable for sexual abuse. Ms. Bash was referring to the defamation lawsuit filed by the writer E. Jean Carroll, which is now on trial.

“This one I haven’t looked at,” Ms. Haley contended, “but if he is found guilty, he needs to pay the price.” (Mr. Trump has already been found liable in a civil trial, and a jury is only weighing how much he has to pay.)

Although she was slow to make inroads in Iowa, she has been heavily competing in New Hampshire from the start, buoyed by a recent influx of cash from her allied super PAC and the prospect of a more independent and college-educated electorate. On Wednesday, her chief opponent for second place, Gov. Ron DeSantis of Florida, announced he would be relocating the majority of his staff to South Carolina, her home turf, while largely pulling back on events in New Hampshire.

Mr. DeSantis is aiming to take advantage in her own backyard ahead of that late-February primary, where the Republican base is more in line with the largely white and Christian evangelical voters found in Iowa. Ms. Haley, for her part, has long contended that she would perform well enough in Iowa and New Hampshire to catapult her campaign into a showdown with Mr. Trump in South Carolina.

But Mr. Trump continues to hold leads in both states. It was in South Carolina where Mr. Trump solidified his path to the G.O.P. nomination seven years ago. And a new poll from New Hampshire’s St. Anselm College shows that while Ms. Haley is handily winning over New Hampshire independents, Mr. Trump’s hold on Republican voters is so overwhelming that he leads by double digits, 52 percent to 38 percent.

Nevertheless, he has responded in kind to Ms. Haley, seeking to fully dispense with his nearest rivals after a blowout victory in Iowa.

At his rally in a country club in Atkinson, N.H., this week, he attacked her track record as governor and portrayed her as out of touch and not Republican. His campaign has sent out a slew of emails with subject lines including: “Nikki Haley Loves China,” “Nikki Haley Is Funded By Democrats, Wall Street, & Globalists” and “Nikki Haley Is Weak On Immigration And Opposes A Border Wall.”

On Wednesday in Portsmouth, N.H., he further polished his attacks on Ms. Haley, pointing to her past positions on taxes, Social Security and Medicare and condemned her for being supported by liberals and moderates.

Citing a poll that showed that nearly half of Ms. Haley’s supporters in Iowa said they’d vote for President Biden over Mr. Trump, the former president said, “Now, that means that she’s like a Democrat. I actually think she might go to the Democratic Party.”

Later, he criticized her as being too weak on border security, a signature issue that has animated Mr. Trump’s presidential campaigns. As he accused her of supporting open borders, he told hundreds in New Hampshire, “I don’t know that she’s a Democrat, but she’s very close. She’s far too close for you.”

Mr. Trump was also set to say that “Nikki Haley wouldn’t just lose the White House, she’d lose us the House and Senate,” according to remarks shared by his campaign. He left that line out of his speech, but it was reported online, and Ms. Haley had a ready rebuttal. While Mr. Trump was still onstage, she wrote on X that Trump “is confused about his own record,” and noted that Republicans lost the House, Senate and presidency during his tenure.

Earlier in the day, her campaign preemptively rebutted his criticism on her record, releasing a two-minute compilation capturing Mr. Trump thanking Ms. Haley for her work as governor and ambassador.

Democrats have been hitting Ms. Haley, too. They have pointed to her signing of strict abortion and immigration restrictions and her careful treatment of Mr. Trump as signs that she is part of the same old wave of extremist Republicans as Mr. Trump, rather than a fresh face.

“No matter how they dress it up, these MAGA Republicans have already told us how they want to crash our economy and rip away our freedoms for their own political gain — and we’ll take them at their word,” Ray Buckley, chairman of the New Hampshire Democratic Party, said in a statement.

Neil Vigdor contributed reporting from Bretton Woods, N.H.

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