If you remove a few treads in the middle of the stairs, that will give you some better access from above. Consider using a post hole shovel for digging out the holes (probably not worth buying, but certainly renting one for an afternoon would suffice). This is a manual shovel specially designed for digging deep holes - it has a hinged blade and two handles.
You could also consider adding some extra supports to the stringers. You could "sister" additional lumber underneath or alongside the stringers, such as a 2x8. Another approach would be adding a brace (or pair of braces) from the center of the stringers, back down to the base of a deck post, if there is one suitably closeby.