Former Sen. Fred Thompson’s (R-Tenn.) exit from the presidential race on Tuesday freed up his almost two dozen Capitol Hill supporters, but there was little evidence among them of a rush to join other campaigns.

Thompson ended his bid after a distant third-place finish in the South Carolina GOP primary on Saturday.

Rep. Zach Wamp (R-Tenn.), one of Thompson’s most vocal advocates on the Hill, said he was disappointed the Tennessean’s campaign didn’t go further, but he asserted that his candidacy still had a major impact on shaping the debate.

“I think he came out of the gate strong, but late,” Wamp said. “He really hit his stride well in South Carolina, but it really was too late.”

Wamp was one of 22 Senators and House Members to endorse Thompson’s campaign. Of those, the endorsements of Senate GOP Conference Chairman Lamar Alexander (Tenn.) and House GOP Conference Chairman Adam Putnam (Fla.) are likely to be the most coveted by other campaigns.

But Alexander said on Tuesday that he plans to hold off on choosing another candidate, at least for now.

The Republican Conference chairman said he has close relationships with several of the remaining contenders and has his hands full with his new role as the No. 3 GOP Senate leader.

“I’m not likely to jump very quickly to endorse,” Alexander said. “There are several good candidates.”

Putnam also has no plans to choose a new candidate before the Florida GOP primary, scheduled for Jan. 29.

“I think it is unlikely that I would endorse before Florida’s primary,” he said Tuesday. “I believed very strongly in Fred Thompson and all that he brought to the table, and I think that it would be very disingenuous of me to switch horses that quickly.”

With Thompson’s announcement, speculation swirled about whether he would now make an endorsement.

Wamp said it was his understanding that the former Senator would remain neutral for the time being and not immediately endorse one of his former foes. Wamp said he too is keeping his powder dry for now.

But a source close to Thompson speaking on condition of anonymity said there was a better than 50 percent chance the former Senator would back Sen. John McCain (R-Ariz.) — with whom he has a long-established friendship dating back to their days in the Senate. A Thompson endorsement of McCain could provide a much-needed boost to the Arizona Senator’s standing with the GOP’s conservative base.

Asked what impact a Thompson endorsement would have on McCain’s candidacy, Alexander said: “I suspect it would reconfigure the race. He has strong conservative appeal.”

Sen. Richard Burr (R-N.C.), a McCain backer, said he believes a Thompson endorsement of McCain would serve as the latest boost to a surging candidacy.

“It wouldn’t surprise me if you see new endorsements every day,” Burr said. “Campaigns are all about momentum and you want to continue the flow of that.”

As far as who gains the most from Thompson’s exit, Wamp said McCain most logically does, but former Arkansas Gov. Mike Huckabee will get a boost, too.

“A lot of Thompson people like Sen. McCain,” he said. But “Huckabee in the South will pick up some of Fred’s votes.”

Wamp also indicated that Thompson would make a strong running mate for the eventual Republican nominee and would provide ideological and geographical balance on a ticket with either McCain or former Massachusetts Gov. Mitt Romney. Other allies also have been floating Thompson as vice presidential material.

“I feel like he would add a whole lot to our ticket going forward,” Wamp said. “He would make a tremendous contribution to our ticket.”

While it is unlikely that most of Thompson’s Congressional supporters will shift their endorsements to other GOP candidates before Feb. 5, when 22 states will hold their primaries, it doesn’t mean they won’t feel the pressure to do so.

“I would speculate they are all getting calls from other camps,” said the Thompson ally.

Sen. Thad Cochran (R-Miss.), who had been a Thompson backer, said Tuesday that he already had switched his support to Romney.

“I’ve been impressed with him all along,” Cochran said. “He has the personal qualities and the experience to be a top-notch president.”

Still other Thompson supporters indicated they plan to stay on the sidelines now.

Rep. Lynn Westmoreland (R-Ga.) said he is staying neutral in the presidential race. In a statement, he acknowledged that Thompson’s campaign never really got off the ground.

“A lot of Republicans, at least in the South, felt like there was something missing in the GOP field. We really wanted Sen. Thompson to fill that gap,” Westmoreland said. “I’ve heard of peaking too early, but this campaign seemed to peak before it even officially started.”

It remains unclear whether any Thompson staff will move over to McCain’s camp.

“I wouldn’t look for a package deal,” said the Thompson ally, suggesting that most of the senior campaign staff will likely “migrate back to similar places from which they came.”

The one exception could be Todd Harris, Thompson’s communications director who worked as McCain’s traveling press secretary during the 2000 presidential campaign. Harris took a leave of absence from the DC Navigators, a lobbying firm, to work for Thompson’s campaign.