MANCHESTER, N.H. — Representative Ron Paul of Texas finished a strong second in the state’s Republican primary on Tuesday, which in many ways was the more telling outcome in a race where Mitt Romney’s dominance was never in doubt.

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Mr. Paul polled well ahead of the late-surging Jon M. Huntsman Jr., who ran third, and Newt Gingrich and Rick Santorum, who battled for fourth. Mr. Paul benefited from the large turnout of independent voters, getting the nod from about a third, a little more than Mr. Romney. He also did well with young voters and those who said they were liberal on social issues.
But even if political analysts continue to regard the libertarian-leaning Mr. Paul as a protest candidate, with no shot at the nomination, his success here — on top of a third-place finish last week in the Iowa caucuses — means he will probably continue his campaign for months and perhaps to the summer convention.
“There is no way they are going to stop the momentum that we have started,” Mr. Paul told a raucous crowd shortly after Mr. Romney gave his own speech.
He bragged about getting the political system to talk about “real cuts” in spending, monetary policy and the Federal Reserve.
Despite Mr. Paul’s strong showing, it is clear that he will have a hard time repeating that success, at least in the next two primaries.
In South Carolina, which votes on Jan. 21, Mr. Paul has a reasonably strong organization but one that is not considered as powerful as in Iowa and New Hampshire. He also faces heightened opposition, some party strategists say, because of his noninterventionist and antiwar positions. South Carolina has a legacy of large military installations and heavy defense industry and military employment.
After South Carolina comes the Florida primary on Jan. 31, the first big-state contest. But Mr. Paul is largely expected to bypass that race because of the huge expense of television advertising and other campaign costs, as well as the structure of the primary, which is seen as putting him at a disadvantage.
Instead, he will focus on the Nevada caucuses on Feb. 4 and on caucuses in other states where he has worked to build organizations and where the rules make it easier for independents to vote for him.
The other story in New Hampshire was Mr. Huntsman, who did much better here than the single digits he showed in national polls, benefiting from his sole focus on the state.
But with the Republican nominating contest moving to South Carolina, Mr. Huntsman faces a steep challenge, with little organization in the state and no personal connection to the voters. His moderate views also mean that he has an uphill battle in a Southern state with many conservative Christian voters and Tea Party members.
Over all, Mr. Huntsman performed best here among voters who either support the Obama administration or share some of its views, according to exit surveys.
South Carolina’s demographics, on the other hand, will favor the two candidates battling for fourth here, Mr. Gingrich and Mr. Santorum.
Each of those candidates will try to prevent South Carolinians from accepting the inevitability of a coronation of the New Hampshire winner, which they performed in 2008 when Senator John McCain’s momentum from winning New Hampshire catapulted him ahead of a more conservative candidate, Mike Huckabee, and led to his nomination.
Mr. Gingrich, who believes his one-time lead in national polls was undone by attack ads from Romney supporters, heads to South Carolina with a “super PAC” pledging $3.4 million for attack ads. They will focus on Mr. Romney’s career as a corporate buyout specialist, a case Mr. Gingrich is making with voters and in interviews.
“As we get to South Carolina, as the choice becomes clear,” Mr. Gingrich said, “I believe we can reach out and we can create a majority that will shock the country and a majority that will continue to put us on the right track.”
“It is doable. It is a daunting challenge. But consider the alternative,” he said Tuesday night.
Mr. Gingrich also signaled that he would defend his right flank in South Carolina from the conservative alternative of Mr. Santorum, whom he has called a “junior partner” of his in Congress in the 1990s and who has had less success campaigning.
“I actually know how to build a nationwide campaign, and he lost Pennsylvania by the largest margin of any senator in the history of the state,” Mr. Gingrich said.
After a disappointing showing in New Hampshire, Mr. Santorum is banking on South Carolina to regain the momentum he had when he nearly won the Iowa caucuses.
“We are going to go on to South Carolina,” he told supporters in New Hampshire, where he would be the “true conservative” in the race. His campaign has raised $3 million since Iowa and is already pumping $1.5 million of it into TV ads in South Carolina.
Mr. Santorum said he competed in New Hampshire because “we wanted to respect the process” and campaign in every state.
“We didn’t spend a lot of money, but we put our message out there,” he said. “We came where the campaign was and delivered a message not just for New Hampshire but for America.”
He stayed on his conservative message in the Northeast state, which John Braebender, a top adviser to Mr. Santorum, said would make him appealing to South Carolina voters. He may also benefit if there is a backlash to Mr. Gingrich’s turn toward negative campaigning, after he vowed earlier not to criticize fellow Republicans and to focus on “positive solutions.”
The Santorum campaign also has a bigger presence in South Carolina than it had in New Hampshire, and it is starting to organize county chairmen in Florida.

Reporting was contributed by Richard A. Oppel Jr. in Leesville, S.C., and Katharine Q. Seelye, Marjorie Connelly and Michael D. Shear in Manchester.