Part of the challenge facing Ms. Warren is the warm feelings that many black voters have for Mr. Biden after he served two terms as Mr. Obama’s vice president.

“I think it’s really hard for candidates like Elizabeth Warren, Cory Booker, Kamala Harris, anybody, when you’re going against Joe Biden, who can drop, ‘Obama, Obama,’ and African-Americans are like, ‘Yes, yes,’” said Jazmine Curenton, 19, who came to a block party for Mr. Booker in Columbia.

Mr. Biden’s reputation may be particularly hard for Ms. Warren to overcome among older voters.

“I think that there are a lot of younger people who not only are really open to her, but truly like her and truly like what she’s saying,” said Uchechi Kalu, 26, who has canvassed for Ms. Warren’s campaign and attended her event in Rock Hill. “And then there’s an older generation that is sticking to people who are familiar with the Democratic base.”

Ms. Warren’s team is counting on her standing to improve as she becomes more familiar to black voters.

“What we see is that folks who have an opinion, and have heard of Elizabeth Warren, tend to have a favorable opinion and like her,” Joe Rospars, the Warren campaign’s chief strategist, said in a podcast interview with David Plouffe, who managed Mr. Obama’s 2008 campaign. “What we also see is that people of color are disproportionately likely to not have heard of, and not have an opinion about, Elizabeth Warren.”

Ms. Warren is making a concerted effort to build relationships with black leaders.

“Every couple of weeks, she’ll call just out of nowhere,” said the Rev. Al Sharpton, who has yet to endorse a candidate. When Mr. Sharpton became a grandfather late last year, Ms. Warren was quick to offer congratulations. And last month, when Mr. Sharpton received an award and gave a speech at the annual Congressional Black Caucus Foundation dinner in Washington, he immediately received a text from Ms. Warren.