We had one side of our house jacked up and metal plates (see pic) were installed around the base of the relevant stumps. Then a bolt is passed through the stump attaching it to the metal plates. Most of the old stumps were retained, but some were replaced. We certainly didn't need the whole house restumped, and the house has stayed level in the subsequent 15 years.
I would ask around about the metal plate restumping option. I forget the name of the mob that did ours, but it was very good value (from memory it was two contractors for about three days).
Those are my principles, and if you don't like them . . . well, I have others.