WASHINGTON — For much of the time that Hillary Clinton has run for president, her chief adversary on Capitol Hill has been Representative Trey Gowdy, Republican of South Carolina and chairman of the special committee that investigated the 2012 attacks in Benghazi, Libya.

The committee completed its investigation with a scathing attack on Mrs. Clinton in June, but found no new evidence of wrongdoing. And now she has a new tormentor: Representative Jason Chaffetz of Utah.

The F.B.I. director, James B. Comey, announced on July 5 that the bureau would recommend that Mrs. Clinton not be charged in connection with her use of a private email account when she was secretary of state, but gave a detailed account of what he called her “extremely careless” handling of classified information. House Republicans rejected Mr. Comey’s conclusion, and Mr. Chaffetz, the chairman of the House Oversight and Government Reform Committee, has taken the lead in trying to do something about it.

Just five days after Mr. Comey’s announcement, Mr. Chaffetz asked the Justice Department to open an inquiry into whether Mrs. Clinton had lied in October when she testified before the Benghazi committee. The Republicans’ request has been met with silence from the department and the F.B.I., and prosecutors have shown no indication they are willing to open another investigation. Legal experts have said that making a perjury case against Mrs. Clinton would be difficult.