Mr. Graham, seeking a fourth term, is capturing 89 percent of Republicans, a number he will need to increase to stave off Mr. Harrison’s challenge, given the likelihood of Black voters moving to the Democrat.

The good news for Mr. Graham as it relates to his own base is that he’s now on center stage in Congress, wielding the gavel at the Senate Judiciary Committee each day this week during Judge Amy Coney Barrett’s televised confirmation hearings.

Fifty-two percent of South Carolinians said they backed Ms. Coney Barrett’s confirmation while only 30 percent said they opposed it. The support for her was even higher among college-educated white voters, with 59 percent of them backing her appointment.

Notably, Mr. Graham fared better in the survey in the final two nights of polling calls to voters, after the Supreme Court hearings had begun. In the initial calls, starting last week, the Senate race appeared closely divided. But this week, Mr. Graham actually fared better among respondents than Mr. Trump.