WASHINGTON — As many of his Republican colleagues were sliding into disconsolate resignation, Senator Ben Sasse of Nebraska sat on a riverbank recently and wrote a meandering missive calling on someone to challenge his party’s presumed nominee, Donald J. Trump, for the presidency.

“Why shouldn’t America draft an honest leader?” he wrote on Facebook. “You know ... an adult?”

Mr. Sasse, 44, a baby-faced former college president who outsmarted a bevy of better-known Republicans to win his seat in 2014, has stood out largely for his conservative voting record, and perhaps for his children, who are home-schooled half of the time in Senate committee hearing rooms and on his office couch.

But Mr. Sasse is now best known for his strident — and counterintuitive — opposition to his party’s standard-bearer, one of the last remnants of “Never Trump” in a Capitol slipping into the vortex of the mogul’s general election campaign.

The House speaker, Paul D. Ryan, has not given Mr. Trump an endorsement — notwithstanding occasional unfounded claims by the candidate’s aides that one is coming — but the two men continue to have what the speaker has called “productive conversations” on policy issues. Senators Lindsey Graham of South Carolina and Jeff Flake of Arizona, both Republicans, have treated Mr. Trump like an insufferable relative at a family wedding. But Mr. Graham said he recently had a “cordial” phone call with Mr. Trump on foreign policy, and Mr. Flake has said he is still deciding whether to endorse the businessman.