Mrs. Clinton used two of her most high-profile speeches, including one before a black sorority convention, to address minority voting rights — an explosive issue among African-Americans since the Supreme Court struck down part of the Voting Rights Act in June. A month after Mr. Gray’s funeral, Mrs. Clinton and her husband both asked to speak at the service for Bill Lynch, a black political strategist who is credited with the election of David N. Dinkins as mayor of New York, and stayed for well over two hours in a crowd full of well-connected mourners. And there have been constant personal gestures, especially by the former president.

“I think that this is an effort to repair whatever damage they felt may have been done in ’08,” said the Rev. Al Sharpton. Mr. and Mrs. Clinton “know that there are some who have lingering questions, if not antipathy, towards them,” Mr. Sharpton said.

This task has taken on new urgency given the Democratic Party’s push to the left, away from the centrist politics with which the Clintons are identified. Strong support from black voters could serve as a bulwark for Mrs. Clinton against a liberal primary challenge should she decide to run for president in 2016. It would be difficult for a progressive candidate, such as Senator Elizabeth Warren of Massachusetts, to rise if the former first lady takes back the black voters she lost to Mr. Obama and retains the blue-collar white voters who flocked to her.

Her appearance before the sisters of Delta Sigma Theta in July, which she opened by offering condolences to the family of Trayvon Martin, the 17-year-old who was killed in Florida last year, and her voting rights address to the American Bar Association in August drew significant attention among black leaders.