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.