Ralph Norman

The two-week runoff campaign to decide which Republican candidate will compete in November to succeed the late Senator Lindsey Graham started in the wee hours of Wednesday morning, as each candidate workshopped what their sales pitch should be.

Senator Darline Graham, Mr. Graham’s sister, had holed up at her campaign headquarters in Columbia, S.C., according to two people with knowledge of her night who requested anonymity to describe a private gathering. She discussed next steps on the phone with President Trump while she snacked on chicken nuggets and sweet tea from Chick-fil-A.

Representative Ralph Norman, the vice chairman of the House Freedom Caucus in his fifth term in Congress, was similarly ensconced on the other side of the state. His team released a television ad on Wednesday making the case that he is the only proven conservative in the race.

“You shouldn’t have to guess where your next senator stands,” he says in the ad. “With me, you already know.”

Ms. Graham and Mr. Norman emerged from a special election sprint on Tuesday to succeed Mr. Graham, who died unexpectedly in July. Now the two candidates are jumping into a two-week campaign for an Aug. 25 runoff that will be an even faster-paced foot race.

They defeated a wide field of other candidates in Tuesday’s primary, including Representative Russell Fry and former Gov. Mark Sanford. Ms. Graham is expected to make the case to voters who cast ballots for Mr. Fry, also a close ally of Mr. Trump, that they should choose her if they want a senator who is more aligned with the president. Mr. Sanford and Mr. Norman share a focus on fiscal conservatism.

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