I know that I've seen recipes on TV for hamburgers that included some form of slow cooked meat that had been chopped/ground down and added to the mix, but I've never gone to that much trouble myself.
As temperature is the main issue with how far something is cooked (as that affects how the proteins tighten up and squeeze out moisture), so long as the original meat is fully cooked, you just need to cook the raw meat until it's to a safe temperature.
Of course, in terms of storage safety, you'll want to consider the cumulative time that the older meat has been around, so I wouldn't necessarily consider doing this with cooked meat that's been sitting in my fridge for multiple days if I was going to be serving this to other people (I'm not so risk adverse when I'm just cooking for myself), or treat this as a way to extend the life of the older meat.
But as Anastasia hinted at, you could also mince up the cooked meat and use it other dishes ... and those don't necessarily need to include more raw meat. I've chopped up roasts or steaks to make stews, fajitas, dirty rice, stir-fry, stroganof, casseroles, hash, etc. (although how heavily you've flavored the original meat may affect what you can do with it afterwards)