Republicans are deeply motivated to get a major tax bill passed and signed by the president and have returned to committee chambers — rather than the private back rooms of leaders’ offices — to get the process rolling.

Perhaps most important, Senate Republicans have begun to stiffen their spines against Mr. Trump, who has spent the better part of his presidency alternatively ignoring, undermining or outright denouncing the efforts of Congress to legislate. On Thursday, before leaving on a monthlong recess, the Senate set up a system to prevent the president from appointing senior administration officials to posts that require confirmation in the senators’ absence.

Among its more notable successes this year, and against Mr. Trump’s objections, Congress passed a tough Russia sanctions bill with a veto-proof majority, which the president begrudgingly signed this week. Congress also approved a law to help veterans get health care — a bipartisan, bicameral, messy but ultimately successful effort that came together with zero involvement from the administration.

A complicated debt ceiling fight may be averted now that Mr. Trump’s budget director, Mick Mulvaney, said on Thursday that Congress could lift the ceiling on the nation’s debts without having to make spending cuts in exchange.

“There is more good happening here than people know about,” said Senator Bob Corker, Republican of Tennessee. He added that he expected further bipartisan agreement on various policy efforts now that the Senate has dropped the health care battle and Republicans are gaining momentum on a tax package that they desperately need to win. As for the role of the president in all that, “I haven’t thought about it,” he said.