A cat on a leash? LOL. Actually tried that with one of my escaped pet rats and our younger cat found the general area he was in.

If he's climbing into other cages then maybe set up and easy to get into empty cage with food and whatnot and see if he ends up back in it. If you can shut him in it, cool. If the food goes missing, try putting one of the traps in next to the food dish. Other than that some places have kill traps which actually electrocute the rat/mouse, no escaping from those.