You're going to get a lot of different takes on how to handle the situation but you ask so I'll tell you what I do.
First off, did you do anything when you felt the fish? I mean did you react or did you stay cool and allow the fly to finish the swing and then hang?
Me? I don't react at all. That takes time to develop but I believe that when you react and try to strike back you will put the fish off.
I stand right in my boot prints and take a careful accounting of everything I can. Things like where I think the fly was when I felt the fish. Then consider and land mark on either shore for a more precise way to mark the area. Next I'm looking not at where I think the fish was when it played with the fly but where the fly passed prior to the bump. It is the back track of the flies drift and swing that will tell me where the fish likely was hanging out when it spotted the fly. Any surface disturbances get duly noted then…………
I'll take a walk upstream, not 5 yards but more like 30. Then begin working back down with the same fly keeping a wary eye on where it is I am headed to. The extra time it will take for me to get back down there pays 2 dividends; one is it allows me to get back into a good rhythm of controlling the drift and swing for the purpose of depth & speed control. Two is that the time will give the fish time to re-acclimate and get back to its holding spot which they often seem to do...………
As you get closer you want to be looking at the shore on either side and the water to be sure of where you want the casts to land and the fly to travel. Go slowly because you want this fish to see the fly and only the fly not the leader or sink tip because you rushed things and are fishing to long with the fly passing behind the fish. If I do everything like I just described I raise about 75% of them a second time. If I went upstream far enough and came down slow enough it's been at least 20 minutes and my chances are better than most. I don't change the fly. The fish went for it once and there's no reason for me to believe it won't go again.
All of that hinges on one thing, that you did not react when you felt the fish by jerking the rod back. When you do that to a fish who is investigating something which appears strange but may be elidable you alarm the fish maybe even prick it a tiny bit and if you've done that you may as well move on and find another. If you find another than don't react if you feel a bump but no pull.
Ard
PS. I'd appreciate comments as to whether or not what I have said resonates with others. When I answer a question like this it isn't about being right. It's about trying to share things that took years and repeated experimentation in order for me to develop patterns for the way I do things. And I only write about things that seem to work time after time