Normally, I'd say that at the end of June, hibernation is a bit unlikely... but with the low temperatures this year, who knows...
If you're absolutely sure it's old age hammy is suffering from and a trip to the vet is only going to distress him, just make sure he's warm and comfy in his cage and let him be. Animals don't have the same awareness of illness or approaching death that we do so as long as he feels he is in a dark safe place, it is (imo) one of those rare occasions that you can let nature take its course. As to how long that course is... I don't know. Perhaps a call to your vet?
Hope hammy has a kind and gentle departure.