Mr. Hagel struggled to fit in with Mr. Obama’s close circle and was viewed as never gaining traction in the administration after a bruising confirmation fight among his old Senate colleagues, during which he was criticized for seeming tentative in his responses to sharp questions.

He never really shed that pall after arriving at the Pentagon, and in the past few months he has largely ceded the stage to the chairman of the Joint Chiefs of Staff, Gen. Martin E. Dempsey, who officials said initially won the confidence of Mr. Obama with his recommendation of military action against the Islamic State.