Mike Pence returned to Capitol Hill on Thursday as the vice president-elect, serving as wingman in helping navigate President-elect Donald Trump through the halls of power.

The Indiana governor and former congressman took a leap of faith when he accepted Mr. Trump’s invitation to be his running mate. Now conservatives say that as the experienced Capitol Hill hand serving a president who has never held elected office before, Mr. Pence is positioned to be one of the most influential vice presidents in history.

“Donald Trump comes to Washington as the complete outsider who has never run for office before, who has never worked in Washington, who doesn’t know how to navigate these waters. Mike Pence is going to be that person who does it for him in Washington, D.C. He is the one with the experience,” said L. Brent Bozell, founder of the Media Research Center.

Mr. Pence served in the House for 12 years, earning his conservative reputation by opposing the Bush administration’s No Child Left Behind education law, the Medicare prescription drug benefit, ballooning deficits and the 2008 bailout of Wall Street.

He served as chairman of the conservative Republican Study Committee and then did a stint in House leadership as chairman of the Republican caucus, building ties with Rep. Paul D. Ryan, Wisconsin Republican, and Sen. Mitch McConnell, Kentucky Republican, now their respective chambers’ leaders.

Mr. McConnell said he envisions Mr. Pence serving as a conduit between the White House and members of Congress in much the same fashion that Dick Cheney did for President George W. Bush.

“I’ve mentioned that to Vice President Pence,” Mr. McConnell said. “I hope he will attend our Tuesday policy lunches when he’s in town and kind of be our liaison between the administration and the Senate, much like Vice President Cheney was.”

Mr. Pence was at Mr. Trump’s side Thursday as the president-elect held separate meetings with Mr. Ryan and Mr. McConnell, but he didn’t address the press.

He also met with Vice President Joseph R. Biden while President Obama was showing Mr. Trump around the Oval Office and first lady Michelle Obama was meeting Melania Trump.

As vice president, Mr. Pence will have an office in the Senate, where the Constitution makes him the presiding officer. Vice presidents rarely perform that role, except in ceremonial duties or to break tie votes, but it creates an opening for the kind of bridge-building at which Mr. Cheney and Mr. Biden, a former senator himself, have excelled.

Mr. McConnell mentioned brokering several major deals with Mr. Biden in recent years.

During the campaign, Mr. Trump’s selection of Mr. Pence as his running mate helped assuage the concerns of conservative and Republican leaders who were wary of the brash billionaire. Mr. Pence was even deployed to Capitol Hill to assure Republican lawmakers that the similarities they had with Mr. Trump’s agenda far outweighed their differences.

David McIntosh, a former member of Congress from Indiana who now serves as president of the Club for Growth, which has been critical of Mr. Trump and his comments on free trade, said he believes Mr. Pence will “put definition around Trump’s vision” — a role he also played in the primary race.

“Trump said, ‘I am going to make America great again.’ Mike would explain how free market economics, how conservative justices, how an agenda that truly is small government and pro-freedom will make America great again,’” Mr. McIntosh said.

“I think he will be there as a steady, faithful partner to Donald Trump, bring his experience from being a House member and as a governor into the room and share with him, ‘Here is how we really make work the agenda for making America great again,’ and that is why we thought it was a great pick,” he said. “He is a humble man, strong in his own faith, but also knows what he believes in, and I think that will provide a good, steady rudder for the upcoming Trump administration.”

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