So many whodunnits from the studio era, especially the ones that are more comedy than mystery, have rather arbitrary solutions. At the end, the detective rattles off a bunch of facts that "solve the crime" (possibly with holes filled in by the murderer's confession). However, they could just as easily have rattled off other facts that would point to someone else. The movie Clue, with its multiple endings, satirizes this.
So it's nice to see that this film, despite some pardonable weaknesses in credibility, does show a clue early on that points unambiguously to the real killer. You have to watch the early scenes carefully to get the clue, which I won't reveal here. However, here's a hint: The real murder on top of the fake one was premeditated, and no coincidence.