Two months before he was pushed out as defense secretary, Chuck Hagel penned a private letter to the White House, arguing for new measures to rein in Russian President Vladimir Putin and greater efforts to reassure anxious European allies, according to officials briefed on the matter.

Shortly after the September letter, he wrote another memo calling for the administration to clarify its approach to the conflict in Syria. The two messages capped a year of frustrations for Mr. Hagel, who repeatedly found fault with what he saw...