Fed chairs typically step down after their four-year terms, unless reappointed by the White House, in an effort to ensure as smooth a leadership transition as possible. The last time a chair stayed on past the end of their term was in 1948, when Marriner Eccles continued to serve as a governor for three years. But for Mr. Powell, the current moment called for him to break with tradition.

Mr. Powell’s decision reflects his mounting concerns about the Fed’s ability to operate independently amid increasingly aggressive efforts by the administration to test the guardrails that have long protected that autonomy.

Most recently, the Justice Department began a criminal investigation into the Fed’s renovations of its headquarters in Washington and whether Mr. Powell lied to Congress about the plans. It followed on the heels of the president trying to oust Lisa D. Cook, a governor, over unsubstantiated allegations in August of mortgage fraud. The Supreme Court is currently weighing her lawsuit against Mr. Trump.

For Mr. Powell, the Justice Department’s investigation was just another attempt to browbeat the Fed into lowering interest rates, which Mr. Trump has long demanded the central bank deliver. Weeks ago, Mr. Powell stipulated that he would not leave the Fed until the investigation was “well and truly over, with transparency and finality.” He said his decision would ultimately rest on what he thought was “best for the institution and for the people we serve.”

On Wednesday, Mr. Powell made clear those conditions had not been met despite the Justice Department’s announcement on Friday that it would drop its investigation.