Once had this same problem one summer. Fleas on all three cats and the dog, kept coming back even after our own bug bomb, so when I realized that hadn't worked (fun times since I was alone for a week and had to deal with it myself), I did some quick research and realized the eggs were the issue. Flea eggs are apparently quite hardy and can survive bug bombs, so that's how they kept coming back. What I did that worked was quarantine the pets, throw out a lot of the old blankets/scratching posts/other perches our pets used frequently, and spray down parts of the house myself. Also did a lot of vacuuming to try and get any eggs out of our rugs, which might have been easier for me to do than it would be for you since outside of these rugs we have exclusively hardwood and tile floors. We then got each of them flea collars, which was expensive to do for all four but is doable for a single pet.
I can't remember if we did a follow up bug bomb but I don't think we did. I know we had to backyard sprayed more thoroughly because of the dog, but I'm not sure if you need to do any outside treatment since I don't know if your cat is indoors only or not.