Far from chastened by the debt debate, tea partyers and conservative groups signaled Thursday they’ve concluded they didn’t lose, but rather were sabotaged from within by weak Republicans — and they took the first steps to oust one of them.

Mississippi state Sen. Chris McDaniel announced he would challenge U.S. Sen. Thad Cochran in the Republican primary next year, a day after the GOP’s senior senator voted to end the 16-day government shutdown and grant President Obama more borrowing authority.

Mr. McDaniel immediately saw a flood of support from the outside groups that had rallied against this week’s debt and spending agreement.

“Our country can’t afford any more bad votes that stem from old friends and back-room deals,” said Daniel Horowitz, deputy political director of the Madison Project. “And as witnessed from the recent budget battle against Obamacare, we cannot win against Democrats if we don’t grow our conservative bench in the Senate.”

For the past two weeks, the deep divisions within the GOP have been on very public display.

Sens. Ted Cruz of Texas, Mike Lee of Utah and their allies pushed to withhold all government spending unless Mr. Obama agreed to cancel Obamacare, while party leaders called that a losing strategy and tried to come up with alternatives.