Two O'Clock Courage (movie, 1945)

After nearly running over him with her cab, Patty Mitchell picks up a fare who claims to have amnesia. As he fumbles to remember the basic facts of his identity, Patty becomes interested in the stranger and decides to help him in his search. But as the pieces of the puzzle begin to fall into place, and Patty's interest becomes more personal, the stranger finds that he is the prime suspect in a murder case.