this may be a crazy idea, but why not pour your strip footing in sections and excavate under the existing mono slab? Essentially, dig from the outside down to the bottom of the existing mono slab and keep going till you go down to your local frost level. Do this for a 10' width or so without any cars in the garage and without snow on the roof. Use the soil below the slab as your back form (or put up plywood forms that will be left in place), throw in some rebar along the length of the beam and allow it to stick out past your pour length a couple feet, put up a form a foot or two away from the face of your existing slab edge (to allow concrete to be poured and consolidated), then pour concrete in down to the bottom of your existing slab edge, finally backfill with sand once the forms are removed. I mention doing this in 10' lengths or so, as to minimize the length the slab is hanging. This assumes it is a fairly small garage and not some 40' wide garage with huge trusses. The only challenge would be doing this under the garage door if you have a poured concrete drive. Below is a picture of what I describe. It might be labor intensive, but should be less than jacking the garage up and saw cutting your slab.
Legend for the picture: green is soil line, red is the forms and kicker brace, purple hatch is new concrete footer limits, red circles are rebar, light blue is existing slab, and brown is finished grade after backfilling.