There are a lot of cat and dog rescues that pair older pets with senior citizens. Some of them cover veterinary care for the pet and they will take the pet back if things don't work out, you can no longer care for it, or in the even of the person's death.

This is a wonderful win/win arrangement for older pets that get overlooked by adopters and for senior citizens who want the love and companionship a pet can offer but have no one to care for the pet if they get sick or die.