PEORIA, Ill. — Senator Ted Cruz, who rode a wave of voter fury to Tea Party stardom and once called the Republican majority leader a liar on the Senate floor, has begun appealing to “our better angels.”

Less than three years after colleagues accused him of misleading his own supporters about the efficacy of a government shutdown, Mr. Cruz recently scolded Donald J. Trump for “taking advantage” of “low-information” voters.

And while some of his allies took delight in Mr. Cruz’s lack of endorsements from fellow senators, arguing that this solidified his standing as a Washington outsider, he has begun openly courting establishment figures with a practical pitch: I am your last, best hope.

“Come on in,” Mr. Cruz told a crowd of thousands on Sunday in North Carolina, inviting supporters of other candidates to join him. “The water’s fine.”