Both Nevada and South Carolina have mobilized key supporters.

Gwen Mills, the president of UNITE HERE, a major service workers union, has been calling members of the rules committee to lobby for Nevada, where the culinary union affiliated with her union is influential.

“It skews heavily working-class,” Ms. Mills said of the state. “You’ve got to talk bread-and-butter economic issues in Nevada in order to be able to win, and that’s important for the Democrats nationally.”