Here’s a look at how the others besides Mr. Graham and Ms. Collins figure to be at the center of activity.

Mr. Alexander: As chairman of the Senate health committee, he will be pivotal in any action Congress takes to overturn the Affordable Care Act, and he has urged caution to his colleagues. He has forged a close relationship with Senator Chuck Schumer of New York, the new Democratic leader, and has worked well with Senator Patty Murray of Washington, the senior Democrat on the committee and now No. 3 in her party’s leadership. He left the Republican leadership a few years ago because he felt constrained by a role requiring party-line allegiance.

Mr. McCain: He has already made clear that he will oppose any effort by the Trump administration to reinstate interrogation methods, like waterboarding, that have been deemed to be torture. Given his distrust of President Vladimir V. Putin of Russia, he, along with Mr. Graham, will serve as a check on efforts to foster closer ties with Russia. As chairman of the Armed Services Committee, he will wield tremendous influence over Pentagon policy. Newly re-elected at age 80, Mr. McCain most likely ran his last race, freeing him from electoral concerns about a backlash from the right.

Ms. Murkowski: As chairwoman of the energy committee, she is a strong advocate of domestic oil and gas production, but has also raised concerns about man-made climate change and its increasing impact on her state’s environment. Like Ms. Collins, she has a voting record in support of abortion rights, and she can be more of a libertarian than a conservative. Perhaps most important, she won re-election in 2010 as a write-in candidate and clashed with the party leadership, making her something of a free agent.

Mr. Flake: One of the most outspoken Trump foes in the Senate, he took Mr. Trump on directly at a private party meeting. Both in the House and the Senate, Mr. Flake has challenged his leadership, and in some cases has won, notably on his crusade against the home-state projects known as earmarks. A champion of immigration reform, he is up for re-election in 2018 and is likely to be hit from right and left.

Mr. Paul: He has already threatened to filibuster Mr. Trump’s cabinet picks, and he previously raised the alarm about the reach of government surveillance programs, which could put him at odds with the new administration. He has urged restraint with American military power, putting him distinctly at odds with Mr. McCain and Mr. Graham in that area.