Going into the nationwide contests, Mrs. Clinton held a delegate lead of 91 to 65 over Mr. Sanders. About 880 of the 4,765 total delegates were at stake on Tuesday; under party rules, they will be awarded proportionally based on vote tallies for each of the candidates, with Democratic-leaning congressional districts and areas assigned the most delegates.

Mrs. Clinton was set to win at least 150 more delegates than Mr. Sanders from Tuesday’s states; the final delegate allocation will be determined in the coming days. That outcome would give Mrs. Clinton a bigger lead than Mr. Obama eventually established in 2008, which she was unable to overcome. The delegate haul resulted from a broad cross-section of support for Mrs. Clinton: In Alabama, Georgia and Virginia, blacks accounted for more than half the population in some districts, while Hispanics dominated many of the districts in Texas that allocated delegates on Tuesday. Mrs. Clinton had some of her best results in these regions, and also did well in largely white areas of the south.

Mrs. Clinton won about six in 10 white voters in Alabama and Arkansas, and she performed strongly in white rural parts of Virginia, Tennessee and Georgia — results that her advisers highlighted as evidence that she could do well with working-class white voters in Ohio, Michigan and other states. These voters were a key part of her base in 2008 but are now being targeted by Mr. Sanders as well as Mr. Trump.

“We have the makings of a broad-based diverse coalition that could not just power her to the nomination but make for a winning coalition in a general election,” said Brian Fallon, a Clinton spokesman. “But having said that, I think we have room to grow in certain areas.”

Mrs. Clinton herself signaled that she would be more focused on the working-class white voters who have flocked to Mr. Trump. At her Miami rally, she vowed to help lift up “struggling Rust Belt communities and small Appalachian towns that have been hollowed out” by the loss of jobs and opportunities. Exit polls by Edison Research on Tuesday night showed that Mrs. Clinton performed strongly with blacks in Virginia and Hispanics in Texas, the most powerful Democratic forces in those states, and leading with men, women and white voters.

In 2008, by contrast, Mrs. Clinton did well with white voters and had success in some states with Hispanics, but she repeatedly lost the black vote to Mr. Obama and was never able to demonstrate that she could attract a winning coalition of racially and ethnically mixed voters nationwide.