While Senate Republicans say they must recast themselves as fiscally minded Reagan Republicans and not the “big-government conservatives” the public has increasingly seen them as in recent years, there appears to be little consensus in the Conference on what that mythical Republican looks like.

GOP leaders began Wednesday the process of picking up the pieces, with Minority Leader Mitch McConnell (Ky.), Republican Conference Chairman Lamar Alexander (Tenn.) and other leaders talking during the day, aides said. Following those talks, McConnell issued a statement congratulating President-elect Obama on his victory and vowing to work with Democrats in a bipartisan fashion.

“The Republican leadership stands ready to hear his ideas for implementing his campaign promises of cutting taxes, increasing energy security, reducing spending and easing the burden of an immense and growing national debt. On these, and other bipartisan issues, he will find cooperation in the Senate,” McConnell said.

For conservative firebrand and Republican Steering Committee Chairman Jim DeMint (S.C.), the rebranding means a return to old-line conservative values and approaches, and if leadership is unwilling to come along, new leaders might.

“We have got to clean up, reform and rebuild the Republican Party before we can ask Americans to trust us again. This must begin with either a change of command at the highest levels or our current leaders must embrace a bold new direction,” DeMint declared following Tuesday’s electoral defeats for the GOP.

In an interview, he continued his critique of his party’s leadership, warning that “we’ve allowed a few big spenders to rule and ruin our party” and called the GOP’s soul-searching a “crisis of confidence and a crisis of courage.”

But he stopped short of calling for new leaders, saying that “our leadership needs to change direction” and hew to a more conventional conservative ideology.

For Judiciary ranking member Arlen Specter (R-Pa.), changing the party’s image means shedding some of the harder right-leaning inclinations and to adopt a more centrist-based approach.

“I think that the message comes across very loudly and very clearly that we can’t play just to a limited segment of our party. That elections are controlled by independents to a very significant extent and by swing Democrats who are willing to cross party lines and that means having ideas and policies which appeal to them, and this is a country which has traditionally been governed from the center. That’s why the Republican Party has to move to the center,” said Specter, a moderate.

And then there are those who, like Alexander, see the way forward as something of a hybrid approach that draws on the GOP’s traditional principles of fiscal and personal responsibility using a new set of, as Alexander terms it, “solutions” tailored for the 21st century.

“We simply have been a failure at taking those principles and turning them into solutions more than half the people can support,” Alexander said, adding that Republicans can no longer ride “along on solutions fashioned in the late ’60s, ’70s and ’80s.”

Republicans said McConnell would likely have to tread lightly with the handful of centrists in the GOP Conference.

“Working with moderates is always a better option than trying to force them to do what you want,” said Senate GOP strategist Ron Bonjean, who previously worked for former Majority Leader Trent Lott (R-Miss.). “However, the Republican leadership knows it will need to pick its battles in winning them over. It should be expected that moderates will not always be with them.”

Sen. John Cornyn (R-Texas), who is angling to become the next head of the National Republican Senatorial Committee, said that while “56 is not 60,” he and other leaders are hopeful they can maintain party discipline on big-ticket items. “We’re hopeful we’ll be able to get” Republicans in line, Cornyn said, though he conceded it will be difficult.

“It will be harder. I think it depends on the issue. You will always have a group that votes for their states’ interest,” Cornyn said. “Clearly, we’re going to have a tougher time.”

McConnell will be faced with a slightly more conservative Conference, given the retirements of dealmakers such as Sen. John Warner (Va.) and Sen. Pete Domenici (N.M.).

Despite crushing losses and expected upheaval among House Republican leaders, Senate Republicans have repeatedly insisted that McConnell should not be blamed for the diminished minority and would likely be returned to the leader position during internal elections the week of Nov. 17. Other top Senate GOP leaders are likely to keep their positions as well.

Still, Republicans were in the process Tuesday night of taking stock and talking about the need to return to Republican roots.

NRSC Chairman John Ensign (Nev.) argued that Senate Republicans need to “re-establish what the Republican Party is all about … [and] get back to this big-tent Republican Party” that is united on fiscal conservatism. Although Ensign was not ready to call for a break from socially conservative ideologies, he said issues such as abortion or gay rights should not be at the party’s core.

“I think we lost our way on our fundamentals,” Ensign said, adding that cultural issues “are the issues that we can disagree on as a party.”

The tensions within the Conference will be on display in a number of policy issues, but perhaps nowhere more prominently than when Obama begins the process of nominating judges.

DeMint, noting that Democrats successfully used the chamber to block several of President Bush’s nominees to circuit courts, warned that Obama and Senate Democrats will look to pack the lower courts with liberal judges, which he contends could be disastrous for Republicans.

“They’re going to come back and try to fill those seats. … If Republicans don’t make up their mind to stop this, we’re going to lose ground we can’t make up for generations. I think its going to be a key issue,” DeMint said.

But Specter, who as chairman of the Judiciary Committee helped shepherd Justice Samuel Alito and Chief Justice John Roberts through the nomination process, argued such an approach is dangerous and urged his colleagues to take a more measured approach.

“I wouldn’t want to anticipate any special difficulties in the future,” Specter said, arguing that Republicans should allow the process to play itself out before the GOP decides to block a particular nominee. “Anybody who makes a decision up front to block them is being precipitous. I think we ought to give the new president a chance and see what he does,” Specter said.

Meanwhile, with Sen. Norm Coleman’s (R-Minn.) re-election bid going into a recount and as two other Republicans facing protracted recounts, Cornyn has decided to hold off on resuming his campaign to replace Ensign as the head of the NRSC, a source close to Cornyn said. Coleman also is considering the race. Cornyn’s camp decided early Wednesday to hold off from a full-on campaign for the NRSC chairmanship until the political map is clearer.