So I thought of a solution. Put a swivel hook under the table. It stays out of the way unless someone has a cane, crutch, etc. that needs something to lean against. Then pull out the hook. Now the cane will stay put. When you're done, put the hook back under the table. As long as it's a flat hook, it shouldn't snag anything when not in use. Took me a little while to find something like what I'm thinking of. These are for snow chains, no price listed. This one is a cabinet latch, for $1.06. See how you could put a screw through the hole and the hook would twist around that shaft? Swing it out, swing it back.
It's a really cheap solution that requires an absolute minimum of dexterity to apply. Anyone with a restaurant, a waiting room, etc. could put a cane hook under the tables. Or do it at home with your dining or end table, if somebody in the household uses the kind of mobility aid that is not self-standing. A decent tinker could fiddle around with the design and find something that would work to support a variety of items. If you want to get fancy, ask a blacksmith or a silversmith to make you one with scrollwork. No reason it has to be boring if you've got the money to spend. But if you just want to stop dropping your cane, a buck or two at the hardware store should fix that. If your hands don't work great, you might need help screwing it on, but most people can work a screwdriver.