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Although Mr. Smith topped the June 8 primary results, Mr. Thurmond has garnered the endorsements of several of the other candidates in the race, including Carroll Campbell III, the son of another former South Carolina Republican governor who finished third in the nine-candidate contest.

Over the past two decades, an influx of northerners has turned the region’s politics in a more moderate direction, but the district has still been in Republican hands for 30 years.

The National Republican Congressional Committee has not backed either candidate, but spokesman Andy Sere said some of its leaders have given money to Mr. Scott. House Republican Whip Eric Cantor of Virginia has endorsed him.

“I think everyone in our party recognizes the importance of making sure that we have leaders in the party who are representative of Americans from all walks of life and that’s not limited to race,” Mr. Sere said.

Merle Black, a political scientist from Emory University in Atlanta, said that usually runoffs mean fewer voters because those whose candidates are ousted earlier sometimes sit them out.

“But in this instance where you have an African-American running and you have the son of a former U.S. senator - a legendary senator - those factors may make this different,” he said.