Bless you for taking in this old guy. I took in an elderly Siamese cat last summer, who can't see well and is deaf but nevertheless was abandoned to fend for himself. He was outwardly calm but the stress of being abandoned led to some intestinal troubles such as vomiting and diarrhea. This mostly resolved in about a month but occasionally he still has a bout of diarrhea.
Do you have other pets? I wonder if your cat is like mine, and is feeling stressed about something, and that's causing the intestinal trouble. Since you've had him 5 months I'm thinking the stress could be something other than the abandonment.
Your cat may be in the "gray zone" for hyper-t. The gray zone is a normal reading for a younger cat but evidence of hyper-t in an older cat. You may want to do extended T4 testing.
My hyperthyroid cat did not have the hallmark symptoms of eating well but losing weight. In fact she didn't have any outward symptoms, and hyperthyroid was diagnosed through bloodwork.
I have another elderly cat who is not hyper-t, who stopped eating last summer and was so sick I made preliminary arrangements for euthanasia. He lost a lot of weight that he didn't gain back, but otherwise he has recovered. I never did find out what was wrong with him, other than constipation that the vet resolved with an enema.
IBD is a lifelong condition that wouldn't resolve with a steroid shot. It can be controlled with proper diet. Eliminating all fish-flavor food may help, because many cats can't tolerate fish. Feeding grain-free food may help, because many cats can't tolerate grain. Wellness and Evo are brands of both canned and dry grain-free food. Another grain-free food, dry only, is Taste of the Wild.
My suggestion would be to start with dietary changes - eliminate fish flavors and feed grain-free food. Canned food is better for elderly cats because of the higher water content. If you don't feed exclusively canned, try one or two servings of canned with supplemental dry.