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.

On Thursday, the GOP accepted defeat, passing a bill that gives Mr. Obama a “clean” bill to reopen government through January, and to raise debt through at least February.

A majority of Republicans supported the deal in the Senate, but the situation was reversed in the House, where Cruz allies refused to sign off on a series of plans put forward by GOP leaders to end the stalemate.

Ahead of the vote, each side blamed the other for squandering leverage.

Mr. Obama, Democratic leaders and even many top Republicans said they hoped the tea party side had learned some political and tactical lessons, particularly in fighting a fight that now appears to have been futile from the start.

But tea party lawmakers say they could have won if all Republicans had stuck together, and the lesson they draw is that they need to oust those who surrendered.

“I want my colleagues to imagine simply that Senate Republicans stood together and said: We support the House Republicans in standing with the American people,” Mr. Cruz said Wednesday on the Senate floor prior to the bill’s passage. “If that had happened, I believe this result would have been very different.”

Mr. Cruz’s battle cry seemed to resonate Thursday across the country with a new generation of tea-party-style candidates who have announced bids to run against veteran GOP lawmakers in 2014.

In South Carolina, Nancy Mace, the first female graduate of The Citadel, lampooned GOP Sen. Lindsey Graham for being the sole Republican in the state’s congressional delegation to support the debt deal.

Another Graham opponent, state Sen. Lee Bright, also blasted the deal: “We didn’t get anything … We surrendered again.”

In Tennessee, state Rep. Joe Carr accused Sen. Lamar Alexander of violating conservative principles by voting to raise the debt ceiling without any accompanying spending cuts.

And in Louisville, Ky., Matt Bevin, a wealthy businessman, running against Senate Minority Leader Mitch McConnell, praised Mr. Cruz as the new face of the conservative movement and blasted Mr. McConnell, who wrote the final debt deal with his Democratic counterpart, Majority Leader Harry Reid.

“This shutdown was completely avoidable if we had real leadership in Washington,” Mr. Bevin said in a Web video. “Instead, we have career politicians like Mitch McConnell and Harry Reid that make decisions based on what is political expediency for them, not on principles and not on what is in the best interest of the American people.”

Former Alaska Gov. Sarah Palin, the party’s 2008 vice president candidate, also weighed in, urging her supporters via Facebook not to “be discouraged by the shenanigans of D.C.’s permanent political class today.”

“We’re going to shake things up in 2014,” Mrs. Palin said. “Rest well tonight, for soon we must focus on important House and Senate races. Let’s start with Kentucky — which happens to be awfully close to South Carolina, Tennessee, and Mississippi — from sea to shining sea we will not give up. We’ve only just begun to fight.”

The tea party, though, could face an uphill battle in the 2014 campaign season.

A new Pew Research Center poll that found the popularity of the tea party — including within Republican ranks — has plummeted in recent months and even more so since 2010 when it helped the GOP take over the House.

Now 49 percent of the public has an unfavorable view of the tea party, compared to 30 percent who hold a favorable view.

“The tea party’s favorability rating has fallen across most groups since June, but the decline has been particularly dramatic among moderate and liberal Republicans,” the poll analysis says. “In the current survey, just 27 percent of moderate and liberal Republicans have a favorable impression of the tea party, down from 46 percent in June.”

Jim Manley, a Democratic strategist, said that the budget stalemate may have weakened the tea party within Congress, though it hasn’t been eliminated.

“What’s interesting is that more and more senators have had it with the antics of Sens. Cruz and Lee,” Mr. Manley said. “The ‘just say no’ crowd is down to only 18 in the Senate Republican caucus, a tremendous change from only a few months ago.”

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