The best tip I ever got about gluing loose tenons in chairs was from a professional furniture repairman. He took an 1/8" drill bit and drilled from the bottom of the rung and aimed for the bottom of the mortise. He used epoxy in a tapered end plastic syringe and wedged it tight in the hole. Then slowly pushed in the epoxy until he saw some appear around the mortise. The epoxy didn't mind the old adhesive in the hole and that was critical. Putting white glue on top of white glue just won't hold.