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MOUNT PLEASANT, S.C. — Gov. John Kasich of Ohio thrived in New Hampshire by appealing to centrist voters. But he insisted Wednesday that he would not be discouraged by South Carolina’s more conservative politics.

“They warned me when I was coming down here, ‘You’re going to South Carolina and they’re really conservative down there,’ ” Mr. Kasich, standing on a chair, told an overflow crowd outside a brick-oven pizzeria before heading inside to hold a town-hall-style meeting. “And I’m like, wait a minute. People are people. We all have the same concerns.”

Mr. Kasich emerged from the New Hampshire primary on Tuesday just as he had hoped. After being overshadowed for much of the campaign, his second-place finish behind Donald J. Trump earned him a long-sought moment in the national spotlight, and gave him newfound credibility as he tries to replenish his campaign account.

In South Carolina, despite its different ideological makeup, Mr. Kasich followed the same message that seemed to resonate in New Hampshire.

At the town hall here, near Charleston, he stressed his positivity and his desire for both political parties to work together, as well as the need to help people “who live in the shadows,” including those with mental illnesses or drug addiction.

“Some people try to call me a liberal because I think we should offer opportunity to everybody, not just to some people, but to everybody,” he said. “And I don’t know how the Republican Party got so far off kilter.”

Given his performance in New Hampshire, Mr. Kasich is likely to draw more pointed attacks from his rivals, and might be tempted to reciprocate. One attendee told Mr. Kasich he was impressed by his positive tone, but wondered whether it would continue, asking, “Are you going to the dark side?”

Mr. Kasich said he would not allow himself to become “a pin cushion or a marshmallow.” But he cited his approach to New Hampshire: “The light overcame the darkness of negative campaigning, and I feel great about it.”

“I’m happier when I’m positive,” he said, while still offering a warning. “I’m not going to sit back and let them pummel me.”

He also noted that life would go on regardless of how his presidential bid fares. “If it doesn’t go well here, next time I’ll come to South Carolina, I’ll come down to play golf, and I’ll be a lot happier doing that.”

Speaking to reporters afterward, Mr. Kasich defended his conservative credentials.

“Who’s more conservative than I am?” he asked, listing some highlights from his record, such as balancing budgets and cutting taxes.

“I think that’s the definition of conservatism,” he said. “And if conservatism has switched from a philosophy of opportunity for all, then we better take a good look at what it means to be a conservative. And look, I’m helping to redefine it right now.”

Mr. Kasich’s performance in New Hampshire seemed to be a draw for some in the crowd.

Cindy Cloutier, a newspaper copy editor and page designer who described herself as a centrist Democrat, said she had been going back and forth between Hillary Clinton and Senator Bernie Sanders, until she saw Mr. Kasich’s primary night speech in New Hampshire.

“Watching Kasich, I thought, ‘Maybe that’s an option, too,’” she said. “Maybe Kasich is somebody who can bring people together. He’s more in the middle. And I think the Democrats are going so far to the left that maybe Kasich is my best hope for finding someone in the middle.”

Her father, Frank Cloutier, a retired engineer and a Republican, said he had liked Senator Marco Rubio of Florida “until he shot himself in the foot” at Saturday’s debate.

Now, he said, he was considering Mr. Kasich, too.