Y'know, Nelson Hayward wasn't too smart. I mean even if Columbo did not dig that bullet out of the wall prior to him returning from voting, setting off a firecracker on the ledge would leave a mess. I mean when ya light a firecracker off, there's always blown-up paper all over the place. (Yes, I know this first hand).
Hayward should have also figured about the phone thing. Why didn't he say he wanted to go to his office simply to take a quick nap or something. Because he was exhausted from all the campaigning.