A commonly cited model is President Ronald Reagan’s decision to enlist former Senator Howard Baker and a team including Kenneth Duberstein. “I just think you’ve got to take that risk and go outside your comfort zone of recycled Obama staff people,” the former Obama official said.

Some turnover is likely. John D. Podesta, the president’s counselor, has resisted requests to stay and is likely to leave after the State of the Union address to head Hillary Rodham Clinton’s presidential campaign. Longtime Obama aides exhausted after eight years of campaigning and governing, including confidants like Dan Pfeiffer and Benjamin J. Rhodes, have thought about whether to move on.

As the president’s advisers map out the next two years, they have focused on three broad categories: agenda items he can advance without Congress, legislation that might emerge from a newfound spirit of compromise with Republicans, and issues that Mr. Obama can promote even without hope of passage as a way to frame the party’s core beliefs heading into 2016.

Beyond immigration, aides said, Mr. Obama is determined to use his power to push for more environmental rules to curb climate change, and they said new Democratic governors could give the president a chance to expand the reach of his health care program in states where Republican governors had resisted. But Mr. Obama has to weigh the consequences of provoking Republicans by using his power to bypass them if he wants to find compromises, too.

“The world is going to look one of several ways on Wednesday morning,” said a senior White House official, who described strategy on the condition of anonymity. “The challenge to date has been the Republican unwillingness to compromise. Will that change after Tuesday’s elections? It’s too early to say. No one is naïvely thinking that the dynamics are going to change easily.”

David Axelrod, a former adviser to Mr. Obama, said the president should make a fresh effort to work with the new Congress despite deep frustrations that led him to bypass legislators this year. “I really think he has to feel out what there’s a willingness to do,” he said. “What he can’t do and won’t do is put his feet up on the desk and cross days off the calendar.”