At the same time, the anti-Sanders camp remains split: Mr. Biden, Senator Elizabeth Warren, Mr. Buttigieg and Ms. Klobuchar are all insisting on staying in the race while Mr. Bloomberg is spending heavily on advertising in Super Tuesday states like Texas and California, where Mr. Sanders could build up a delegate lead if the remaining vote is divided.

In the aftermath of Mr. Sanders’s triumph in Nevada, the question that increasingly looms over the race is whether Super Tuesday, when 16 contests are held on March 3, will be the day he puts himself on a glide path to the nomination.

On Sunday, his aides began taking steps toward trying to do just that.

After months of downplaying Mr. Sanders’s prospects in South Carolina, where he lost badly in the 2016 primary because of his weakness with black voters then, his campaign was blunt about its objective in Saturday’s vote.

“We’re fighting to win in South Carolina,” said Jeff Weaver, a top Sanders strategist, noting that polls show them “moving up considerably on the vice president.”

Should Mr. Sanders defeat or even finish near Mr. Biden in South Carolina, a state the former vice president has long seen as his firewall, it could vault him into Super Tuesday three days later with such force that it may be difficult for any of his opponents to catch up with him.

That’s when Mr. Sanders is hoping to pile up delegates while also inflicting symbolic blows on his opponents by winning in some of their home states, including Ms. Warren’s Massachusetts and Ms. Klobuchar’s Minnesota, both of which Mr. Sanders plans to visit before March 3, according to aides.

Yet it is Mr. Sanders’s decision to compete aggressively in South Carolina, where he has long lowered expectations by vowing only to “do well” there, that illustrates how rapidly this race is moving to him. More than half of the state’s Democratic electorate is expected to be African-American, one of Mr. Biden’s strongest blocs of support, and the former vice president has led in the polls there since the outset of the race.