Now that the Republican presidential primary race has moved to New Hampshire from Iowa, a few things will change.

The evangelical Christian social conservatism that dominates Iowa’s Republican politics is out, replaced by fiscal hawkishness and a libertarian streak rooted in the Granite State’s “Live Free or Die” ethos.

With Iowa fully in the rearview mirror, expect to hear a variation on the phrase “Iowa picks corn, New Hampshire picks presidents,” a favorite local slogan that aggrandizes the state’s role in the nominating process. Still, ask Pat Buchanan and John McCain about how winning New Hampshire in 1996 and 2000 catapulted them to the White House.

One thing is clear: New Hampshire Republicans think their attention to federal spending and the national debt makes them a lot smarter than their Iowa brethren, for whom abortion and transgender issues have been atop the agenda.

“You have a more sophisticated electorate in New Hampshire,” said Ryan Williams, a Republican operative who got his start in the state working for Mitt Romney’s presidential campaigns. “New Hampshire voters exude rugged individualism and flinty Yankee frugality. It’s a different situation.”

Exit polls from Iowa suggested that many of the state’s Republican caucusgoers decided to back former President Donald J. Trump long before the campaign’s final stretch there. New Hampshire voters have a well-earned reputation for making their decisions late. Mr. McCain’s 2008 victory — which did propel him to the nomination — followed a final surge over Mr. Romney and other rivals.

This year, former Gov. Nikki Haley of South Carolina is betting that there’s enough elasticity among New Hampshire voters that momentum can carry her past Mr. Trump and Gov. Ron DeSantis of Florida.

For weeks, she has played to the vanity of New Hampshire Republicans. When she said that the state’s primary voters would “correct” the outcome of Iowa’s caucuses, she was channeling a long-held belief among Granite State Republicans that they have a better grasp on what sort of politician is best suited to becoming the party’s presidential nominee.

Her remark earned her an attack ad from Mr. DeSantis that was in heavy rotation on Iowa television. But the same sentiment is likely to win her plaudits in New Hampshire.

“We don’t know from the polling what’s going to happen next week,” Chris Ager, the chairman of the New Hampshire Republican Party, said Tuesday on CNN. “We just have a very independent-minded electorate, and they’re not going to look at what happened in Iowa and make a decision based on that.”

Mr. DeSantis, who has accused both Mr. Trump and Ms. Haley of being insufficiently devoted to banning abortion, and has placed his opposition to transgender rights at the center of the campaign, has not focused heavily on New Hampshire. He has shifted his post-Iowa resources to South Carolina.

“People aren’t picking their presidential candidate in New Hampshire based on where they stand on transgender rights,” said Steve Duprey, a former McCain aide who was a Republican National Committee member from New Hampshire before being ousted for not being loyal enough to Mr. Trump.

Last year, Mr. Duprey endorsed Senator Tim Scott of South Carolina in the presidential race. Then, when Mr. Scott dropped out, he threw his support behind Ms. Haley, and he is now working to piece together the old McCain coalition to help her stop Mr. Trump from running away with the nomination again.

The challenge Mr. Duprey faces is that for so much of the party, the old focus on fiscal issues has been replaced by the magnetism of Mr. Trump, who in 2016 won the state’s primary election by nearly 20 percentage points. That contest all but crushed the hopes of several moderate rivals, including Jeb Bush, Marco Rubio and Chris Christie, and gave an improbable boost to John Kasich, who finished second.

Now, Ms. Haley’s chances of showing she can compete with Mr. Trump beyond next week may depend on her ability to reincarnate herself as the next coming of Mr. McCain as the campaign moves to her home state, South Carolina. The big question is whether New Hampshire Republicans see her that way.

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