[Porchdog]

Given the appearance of that loose brick I am doubting that there is any other footer. Absent an existing, intact and stable footer I would clear the brick out, Pour a footer below the frost line then put in the new foundation wall.

I wouldn't want to do block in a confined space but I suppose that would be OK. So long as you can form concrete that is relatively easy. My brother and I had a form blow out but we were just doing the "farmer" thing. We didn't screw that up again.

I'm thinking that shoring and jacking the house would be the toughest part. It looks like there are substantial beams under there which would make it easier. If you need to jack on joists you will need beams or timbers to span under them.