After the debate, Mr. O’Rourke and his team held a conference call with donors, telling them he would be more prepared for the next round in July.

But it might not have been enough to appease some major backers.

Some of Mr. O’Rourke’s biggest exclusive donors are now considering giving to other candidates as well, according to interviews with six major donors and those close to them. And in interviews with nearly two dozen undecided voters attending events for Mr. O’Rourke, Sen. Bernie Sanders of Vermont, Sen. Kamala Harris of California and Vice President Joe Biden in Iowa around the Fourth of July holiday, most said they liked the Texan but that they didn’t consider him a top contender.

“He was anointed and then the wax melted,” said Tim Powers, 63, a retired firefighter living in Waterloo, Iowa. “I think he can still come back, but when the light first hits everyone gets shook up.”