Is it possible for a non-professional pilot to pursue a post-retirement career as a professional pilot? Of course I'm not talking about airline jobs but what about bush piloting or missionary/non-profit work?
It depends on your jurisdiction. A lot of places mandate a retirement age of 65 for scheduled commercial pilots (part 121 here in the USA). You are allowed to fly 91 and 135 (charter) operations over 65. While you state you don't want to fly for the airlines, it's worth noting that this retirement age causes a large influx of older out of work pilots many of whom still want to fly. As such there are plenty of high experience older pilots that are primed for flying 135 operations. These may be the people you are competing against for a job which as a low time, potentially inexperienced pilot would be an issue at the bigger outfits. In reality it will depend on a bit on if you are getting into it to make a living or just to fly and bring home some extra cash.
With that in mind there are plenty of commercial operations that seem to always be looking for people:
All of the flight schools near me seem to constantly be on the hunt for instructors. If you are willing to go that route I would think you should have no issue finding work.
There is a small charter op run out of the FBO I fly from. They run 3 King Airs and seem to always be short of qualified pilots. A lot of people use these as time building/jump off points so turn over is high. Presenting yourself as someone willing to stick around, has no interest of going on to the airlines, looks pretty good. There are lots of these smaller outfits, generally at local airports.
It all depends on the kind of flying you’re doing but test you can continue to work professionally as a pilot doing Part 91 or similar work. I have a friend who just retired from UAL as a 787 captain and he now flies an Embraer Phenom 100 on the side.