I'm not a vet, but I do work at a vet clinic and, having seen warbles first-hand, I'd bet that that is what your new kitty is dealing with.
The dark center and what looks to be a protruding lesion definitely makes me immediately think of a warble. The cuterebra is a type of botfly that's larvae enter the skin of dogs, cats, and rodents (among other animals) via an open wound, the nose, or the mouth. The larvae then migrate under the skin and continue their growth cycle. It is a nasty condition, but usually treatable with a good prognosis unless the affected area is near an organ or nerve or if there are an overwhelming number of warbles.
By looking at the pictures, it would seem that the larvae is still inside of the cat. The "hole" you see is so that the larvae can breathe under the surface of the skin. You DO NOT want to even attempt to remove the larvae on your own. It MUST be done by a veterinarian or other qualified individual. If the entire larvae is not removed, it could cause a long-term reaction and if it is ruptured more serious illness (even anaphylaxis) could develop. Since you mentioned another affected area which has cleared up, my guess is that you have a cat with multiple warbles -- some of which have larvae that has left the body. Even these areas will need to be treated by the vet. If larvae is found, it will be removed by the vet and the affected area cleansed and debrided. Your cat will be put on antibiotics to prevent or treat secondary bacterial infections. If the warbles were severe enough, surgery to remove the affected tissue may be recommended. But, you would still be given antibiotics and possibly other medications for use following the vet visit.
It is likely that your cat is also suffering from abscesses as a result of the warbles or from a previous wound. Therefore, regardless of the cause, your kitty needs to be seen by a veterinarian ASAP. While many can go their whole lives without seeing a warble, they are more common than one may think and are transmitted to cats via food sources and areas where rodents, rabbits, and the like are prevalent. Please let us know what happens at the vet appointment (and please let the vet know it is a possible warble or abscess and that it needs to be seen and treated quickly -- be persistant!) and how your kitty is doing afterwards. But, even though we may have an idea of what this is via the internet, a vet is absolutely necessary.
Good luck and please keep us updated. And, also, once your cat is doing better, we'd love to see more pictures and hear more about him or her!
Oh, and welcome to TCS!