Hillary Clinton moved aggressively on Sunday to press her advantage in the presidential race, urging black voters in North Carolina to vote early and punish Republican officeholders for supporting Donald J. Trump, even as Mr. Trump’s party increasingly concedes he is unlikely to recover in the polls.

Aiming to turn her edge over Mr. Trump into an unbreakable lead, Mrs. Clinton has been pleading with core Democratic constituencies to get out and vote in states where balloting has already begun. By running up a lead well in advance of the Nov. 8 election in states like North Carolina and Florida, she could virtually eliminate Mr. Trump’s ability to make a late comeback.

At times, Mrs. Clinton is going beyond seeking simply a victory over Mr. Trump, asking voters to strengthen her hand in Congress and repudiate not just Mr. Trump but also Republicans who have accommodated or endorsed him.

After lashing Senator Patrick J. Toomey of Pennsylvania in a speech on Saturday, Mrs. Clinton urged voters at an outdoor rally on Sunday in Raleigh, N.C., to elect a Democratic governor and to turn Senator Richard M. Burr out of office.