one could dig down with a post hole digger, and under cut the slab some,
I would dig about 4 feet at a time and then fill with concrete, and after it sets dig some more and do again, if one wanted to do more concrete at one time dig a number of areas to pour at one time, If wanted.
by digging a tapered hole wider at the bottom, you will have a "footer" and a stem wall,
one level it off at the bottom of the slab, (fill up to the bottom of the slab), so there is no gaps and take a rod and work it well so it fills the voids,
I have done this type of rebuild on some stem walls foundations, and by doing sections one does not need to worry about losing or sagging the building, as most building swill span 4 foot Ok,
another advantage to this type of system one can make a pad on the out side of the building to jack or support the building from, if your planning on building under the wall you will need to not have direct support under the wall, but supporting it off the sides of it so you can build under it,