The challenge for Mr. Nadler and other House committee leaders is now figuring out how to navigate to secure the material they need for their work. The Judiciary Committee could authorize another subpoena for Mr. Barr’s testimony.

But it then has different paths to escalating its case. Some lawmakers are arguing for opening an impeachment inquiry, which grants the House clearer powers to command information from the executive branch. Others want to hold Mr. Barr in contempt of Congress, a threat that could cajole him into voluntarily giving up more. They could also go to court to try to enforce their subpoenas. Each option has limitations and could slow Congress’s work substantially.

Democrats have still been unable to secure testimony from Mr. Mueller himself. Mr. Nadler said on Wednesday that they were hoping to hold a hearing on May 15, but were still “seeking to firm up the date” with the Justice Department. It is also unclear if Donald F. McGahn II, the former White House counsel whom the committee subpoenaed to testify this month, will show up.