That's a good idea. I was able to work out a solution after my op but it would be nice if Pumpkin's vet offered such an arrangement. I don't think they do because when I took Pumpkin to his checkup yesterday, I gave them a copy of a notarized letter that authorizes a well-known animal rescue group in this area to take Pumpkin should my now-former care manager be unable to do so. The vet didn't say anything about what they do for others like me in the same bind.I think that's part of the answer, right there. Talk to your vet.
Vets are people who care about animals. They are also people who have contacts.
They might know somebody at an animal rescue and adoption group. They might have friends or colleagues who can help. Who knows? Maybe one of the vet techs who works there might want to adopt? I don't know. This is the kind of thing that you would have to ask about.
Call your vet and tell them that you don't have anybody in your family who will care for your cat. Tell them that you don't want to put your cat in a shelter where "just anybody" could take him, assuming that he gets adopted, at all.
Ask the vet if they know of any good rescue or adoption groups that will make sure your cat will go to a good home. Or, maybe if you luck out, somebody who works there might adopt him.
The vet where we take Casper has two cats that live in the clinic and act as greeters. I don't know... that's a long shot but who knows? It might be cool if your cat could be the "Ambassador in Chief" at vet clinic!
I don't know if any of these ideas are doable but you'll never know if you don't ask.
Once you have a plan, put it in your will or, at least, write it down and get it notarized then put it with your important papers.
Remember: Agreements are only as good as the paper they are written on.
I posted that document in a prominent place in my home should something happen to me.