Sen. Thad Cochran waves to supporters as he leaves a campaign stop on June 4. (Rogelio V. Solis/AP)

An army of national tea party groups eager to claim what would be their biggest victory of 2014 vowed Wednesday to intensify their efforts to unseat Sen. Thad Cochran (Miss.) in the next round of the GOP Senate primary.

Wednesday began a three-week runoff campaign that promises to serve as the staging ground for the conservative movement’s most fervent push against targeted Republican incumbents this year. The runoff follows an election Tuesday in which Cochran won 49 percent of the vote and state Sen. Chris McDaniel, who was backed by the tea party, took 49.5 percent. Realtor Thomas Carey had 1.5 percent. Because no candidate won a majority, the race between the top two will go on.

Cochran’s allies promised to wage a full-fledged battle, fueled in part by loyalty to the six-term senator and by fears that McDaniel could put a safe Republican seat at risk.

The result is a sprint toward the June 24 runoff election pitting tea party activists bent on retiring a longtime moderate against establishment GOP forces seeking to avoid a replay of recent elections in which flawed and controversial nominees cost them winnable Senate seats.

Conservative groups have spent more than $5 million on a barrage of attacks against Cochran. Cochran’s allies have spent $2.8 million.

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About half the outside money for McDaniel has come from the anti-tax Club for Growth, which urged Cochran on Wednesday to forfeit.

“He should do the honorable thing and decline to contest the runoff,” Chris Chocola, the group’s president, said in a statement. “Should he choose to persist, the Club for Growth PAC and conservatives throughout Mississippi will vigorously pursue this race to its conclusion.”

The group’s threat was met with disdain by Cochran’s allies, who say they will frame the contest as a battle between outside interests and Mississippi values. Haley Barbour, a former Mississippi governor, was particularly dismissive.

“They’ve bled themselves white trying to get the only scalp left,” he said. “They know if Cochran wins they’re out of business. They will have gone 0 for 2014.”

Barbour’s nephew Henry Barbour runs Mississippi Conservatives, a super PAC designed to buttress Cochran by attacking McDaniel. The group, which has spent $1.7 million doing that, promised Wednesday not to let up. The U.S. Chamber of Commerce, which has spent $500,000 to help Cochran, also promised to stay in the fight.

“We’re not running away from Senator Cochran,” said Rob Engstrom, the Chamber’s political director. “We’re going to continue to stand by him.”

But neither group said how much it was prepared to spend. Another establishment group, the super PAC American Crossroads, said it will not get involved in the race.

At the outset of the runoff campaign, the big question seems to be how pro-Cochran forces will perform against the intensity of pro-McDaniel groups, which appear to be more energized and better funded. Recent history suggests runoffs favor the challenger.

“I definitely think it’s going to be a tough road, even though he’s an incumbent,” acknowledged John Dane III, a yacht company chief executive who contributed money to Mississippi Conservatives.

In Washington, senior Republicans appeared divided over how deeply to wade into the contest with financial backing for Cochran, given that the National Republican Senatorial Committee expects to spend heavily this fall in at least seven battleground states.

Sen. Rob Portman (R-Ohio), a vice chairman of the NRSC, said the committee’s pledge to “fully support” Cochran probably meant a continued coordination among staff members on media strategy and polling, equal to what the Washington-based group has done for the incumbent. The NRSC has not aired expensive TV ads, and Portman suggested that it was unlikely to do so in the next three weeks, leaving that to the Chamber and Mississippi Conservatives.

“I think the big money is going to come from the outside groups,” Portman said.

Other senior Republicans, however, signaled their concern that anything short of a fully engaged effort would leave Cochran disadvantaged and ensure McDaniel a victory. These Republicans fear that McDaniel is a repeat-in-waiting of the 2012 Senate campaign of Richard Mourdock, the Indiana Republican who knocked off 36-year incumbent Richard G. Lugar and went on to lose the general election to Democrat Joe Donnelly.

They worry that some of his positions could alienate the centrist base and mobilize Democrats. McDaniel initially expressed uncertainty about the need for Hurricane Katrina relief; he blamed gun violence in Canada on hip-hop culture, and the Cochran campaign has raised questions about what his team knew about a blogger who was arrested for allegedly filming Cochran’s bedridden wife in a nursing home.

The Democratic Senate nominee in Mississippi is former congressman Travis Childers. Party strategists think his conservative profile — he voted against the Affordable Care Act, for example — would make him the perfect foil for McDaniel.

In an interview, Childers named Donnelly, with whom he served in the House, among the senators he admires. “He was one of those people who befriended me from day one,” Childers said. “Joe Donnelly, I think, is a great senator.”

Democrats see an opportunity to expand the Senate map, even in deeply conservative Mississippi, because of McDaniel’s missteps and potential for errors. John Anzalone, whose firm is polling for Childers and who worked on the Mourdock-Donnelly race, said McDaniel may even be a riper target than Mourdock.

Donnelly’s team “ran a fantastic campaign and they were prepared for when Mourdock made a mistake,” Anzalone said, referring to Mourdock’s remark late in the race that pregnancy resulting from rape is “something God intended to happen.” McDaniel, Anzalone said, is making mistakes now.

One senior Republican, who spoke on the condition of anonymity to be candid, said that a McDaniel-Childers matchup in the fall probably would force the NRSC to spend some of its war chest on a race that otherwise would be crossed off both parties’ lists if Cochran were the nominee.

Cochran will face new challenges in the runoff. His campaign has been cautious and he has declined to debate McDaniel. In a close race, Cochran will have to rally more than his own base.

“I think it’s going to be tough because the 5,000 or so votes that went to the other candidate were possibly votes against Cochran,” said Dane, the yacht company executive, referring to Carey, the third candidate. “There’s a likelihood [those votes] could go to McDaniel.”

Dan Balz contributed to this report.