COLUMBIA, S.C. — With Donald J. Trump's decisive victory in New Hampshire and no strong runner-up among a pack of also-rans, the Republican race barreled into South Carolina on Wednesday shadowed by a question: whether any alternative candidate can gain enough support to threaten Mr. Trump's drive to the nomination.

Gov. John Kasich of Ohio, the second-place finisher in New Hampshire with less than half the support of Mr. Trump, arrives in this more conservative Southern state where he has little staff or support. Senator Ted Cruz of Texas, resuming an effort here to enlist the Christian right, the key to his victory in Iowa, faces a playing field where evangelical voters are far less monolithic. And former Gov. Jeb Bush, buoyed by outperforming his Florida rival Senator Marco Rubio, has a chance to open more daylight — but it is unclear if it will be enough to inspire establishment-leaning Republicans to coalesce behind him.

On the Democratic side, Senator Bernie Sanders's idealistic message, which inspired a decisive victory in New Hampshire over Hillary Clinton, faces a sharp test in South Carolina, where Democrats are more moderate and demographically diverse.

Brochures for Jeb Bush's campaign in the pocket of Collin Thompson, who canvassed for the candidate in Greenville on Tuesday. Credit George Etheredge for The New York Times

Mr. Trump is quite likely to face a kind of scrutiny here he has so far avoided: The only Republican candidate who does not favor increased military spending, he must woo a state with eight bases and 58,000 military retirees. His Vietnam War draft deferments may also be an issue.