South Carolina is not an obvious spot for a potential Democratic upset: The party has not won a Senate race there in more than two decades, it currently holds no statewide elective offices and Mr. Trump is expected to win the state easily. But Democrats are feeling a little momentum that they hope Mr. Harrison can build on. Two years ago, the party was encouraged by the victory of Joe Cunningham, a Democrat, in the race for South Carolina’s First Congressional District, which includes much of the Charleston area.
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Jim Hodges, the last Democrat to be elected as governor in South Carolina, in 1998, said Mr. Harrison’s success, thus far, was in part attributable to college-educated white women in the suburbs of places like Charleston — part of a larger national trend that Republicans are monitoring with trepidation.
“Democrats are winning in suburban legislative races here, and certainly are more competitive in others,” Mr. Hodges said. The “untold story” about South Carolina’s Democratic primary race in February — a crucial momentum-building win for the Democratic presidential nominee, Joseph R. Biden Jr. — “is not just that Biden had strong African-American support, but that about half of the voters in the primary were non-African-American women,” he added.
The state is also growing more diverse, with an influx of newcomers, both foreign and from other parts of the country, who are helping to balloon the population to about 5.1 million today from 3.9 million in 2000 — voters that Mr. Harrison is hoping will turn out to support him in November.
In a phone interview, Mr. Harrison was optimistic that he could persuade some Trump admirers to support him, even though Republicans note that South Carolina has straight-ticket voting that makes such cafeteria-style choices possible but less likely. Some of that support, he said, has to do with the money he’s raised, which has allowed his campaign to share more widely his “Rural Hope Agenda,” including expanding Medicaid under the Affordable Care Act, investing in infrastructure and expanding broadband access in rural communities.
“I know that we have some Donald Trump supporters who are going to vote for Jaime,” he said. “I have talked to them. You know, some of them have come to the rallies because they don’t trust Lindsey.”
Mr. Harrison’s messaging is powered by an assumption that South Carolinians are tired not only of partisanship, but also of the racism and racial divisions that have defined state politics for decades. Mr. Graham replaced the segregationist Senator Strom Thurmond, who retired in 2003. Since then, the state has been a launching pad for high-profile minority Republicans, including former Gov. Nikki Haley, an Indian-American, and Senator Tim Scott, who is Black.