DERRY, N.H. —This is what a political surge looks like:

The men of the Veterans of Foreign Wars hall on Railroad Avenue frantically unfold dozens of extra plastic chairs Wednesday morning, 30 minutes before the arrival of a presidential candidate who barely qualified for the 10-person Republican debate a week earlier. There's a brisk business in bumper stickers for a contender who registered a mere 1 percent in the CBS News Poll less than two weeks ago. Two former New Hampshire senators and a coveted Granite State political strategist circulate among the people who crowded into the hall to hear a White House aspirant who only a month ago had one-third of the support of Donald Trump.

This is John Kasich’s moment.

Here in New Hampshire, site of the first presidential primary, moments like these are no more remarkable than the bracing sound of the lifeguard’s whistle at Beaver Lake a mile from the veterans’ hall or the brilliant colors of autumn at the farm, only two miles away, where Robert Frost moved with his 300 chickens 115 years ago. Sen. Paul Simon had one of these moments in 1988, Sen. Bob Kerrey had one in 1992 and former House Speaker Newt Gingrich had one in 2012. None of them became president, or even finished above third in New Hampshire.

But now the Ohio governor, suddenly in double digits here and running third in the 17-candidate pack — only a point behind former Gov. Jeb Bush of Florida in the Boston Herald/​Franklin Pierce University Poll released only days ago — is having his, and his campaign staff knows that the goal for the next few weeks must be to transform an ephemeral moment into a formidable movement.

The raw materials of that effort filled the chairs in the hall, just off Broadway with its yarn shop, its music store and its breakfast-forever diners. In baseball caps and T-shirts, holding fat handbags and slender handbills, they filed in and signed on, the freshly minted crusaders-for-Kasich. One man, an elected official from a neighboring town, was in shorts that showed off the state's “Live Free Or Die” motto stitched into his black socks. He’s leaning toward Mr. Kasich, too.

To this crowd, Mr. Kasich delivered his unscripted, deeply personal performance, touching, as he did in his congressional and gubernatorial campaigns in Ohio, on his childhood in McKees Rocks and on the lessons he learned from the hard work of his father, who he said “carried mail on his back.” He married these biographical bursts with calls for deficit-reduction progress, entitlement fairness and foreign-policy toughness — a political cocktail swirled with a swizzle-stick of political independence.

“The Republican Party is my vehicle and not my master,” he said, which would be an unusual riposte for a GOP primary but for the fact that Independents can vote here and very likely will hold the margin of victory in the Feb. 9 balloting. “I'm in politics to bring about improvement in society.”

Standing in the crowd were two pros whose allegiance was sought by a dozen Republican candidates.

One, former Sen. John E. Sununu, is part of a family that has been elected to the state House, the governor's office, the U.S. House, the U.S. Senate and, most recently, the New Hampshire Executive Council, an odd institution that, consistent with the Sununu ethos, is a check on political power. He is leading the campaign effort here.

The other, Thomas D. Rath, is a veteran of, among others, the campaigns of Bob Dole, Lamar Alexander, George W. Bush and Mitt Romney.

The base of an isosceles triangle of New Hampshire attorneys general that includes his two direct predecessors, future Sen. Warren Rudman and future Supreme Court justice David Souter, Mr. Rath, who as a Dartmouth student helped organize New Hampshire for Gov. Nelson A. Rockefeller in 1964, is regarded as perhaps the state’s premier political strategist.

His decision to be Mr. Kasich’s state co-chair was big enough news that three television networks rushed stories onto the web.

These men will run Mr. Kasich’s campaign as if he’s running for the state Senate, which he did before he turned 30. That is congruent with the nature of presidential politics here, where voters shop for candidates to support, trying them like the samples at Costco before deciding which is to their taste. That’s what brought former Sen. Gordon Humphrey, a longtime Kasich admirer, to Derry. He wasn’t the only shopper here.

“I’m here to see what he’s like,” said Larry Jordan, a retired Marine. Added Sunne Coleman, a part-time accountant: “I’m evaluating many of these people but I like his fairness, his ability to listen, his vision.”

Mr. Kasich already has a television buy here, but the emphasis will be at the grass roots, with organizations in towns rather than counties. Here in Derry, Mr. Kasich walked easily among voters, asking them questions, engaging their thoughts — not forced but also not seeming excessively forceful in a town-meeting format in a state that holds scores of real town meetings the second Tuesday of every March.

“The field team understands this state,” says Mr. Sununu. “We’ve run here often and we know the arc of a New Hampshire primary — when you do mail, when you do phone calls.”

At the heart of the candidate's appeal is the role Ohio plays; no president has been elected without taking the state since 1960, when Sen. John Kennedy polled 46.7 percent against Vice President Richard Nixon, and no Republican ever has become chief executive without taking Ohio, a streak Abraham Lincoln began when he won 53.2 percent in 1860.

“He can bring Ohio,” says Don Winterton, a town councilor in Hooksett, a 25-minute drive away. “He’s moderate and can attract the independent voter.”

But this is a long campaign — the voting is six months away — and moments like Mr. Kasich’s sometimes truly last only a moment. His strategists regard his competition as Mr. Bush, Sen. Marco Rubio of Florida and Gov. Chris Christie of New Jersey.

The advantage Mr. Kasich — the man of this particular moment — has is that he is having his moment at a time when political allegiances are beginning to congeal here. The danger is that his rivals haven’t had their moment yet, and some of them will.