Supplementing rats with cooked or raw chicken?

So I have an adult snake, and the area that I live in only has a limited availability for food. Often when I go they don't have any mice or rats, or they are way to small. I've supplemented a small cooked piece of chicken in addition to a pinky or hopper because alone it's just not enough food. I've done this twice and my snake was fine. I know this isn't the best idea, but what I was wondering was if it would be ok to feed him a small frozen thawed hen or chick from the supermarket? This would be more of a supplement in addition to a normal diet of mice. Should the meat be cooked or raw? I asked my vet, but he doesn't seem to know anything about snakes. He said he wasn't sure and he also didn't know the proper temperature range to keep him at ( which I know of course, but he had to look it up and it was still wrong) so I don't really trust his judgment. Part of me thinks it should be cooked so he doesn't get sick from salmonella or something like that, but another part thinks that cooked is bad bc he might not have the right kind of enzymes in his stomach to break the food down. As I said before, I have fed him cook chicken twice and he's been ok, but I'm worried that maybe something could build up in his system and get him sick. He'll still be getting the proper nutrition from his other food so this is more supplemental. Any suggestions. Thanks