Let's say I have a hole like this (not my actual hole just one that I found for my situation)
As you can see the existing drywall is flush with the 2x4. What I want to do is cleanly cut away enough of the existing drywall (about an inch) on both sides so I can put in a new piece to fill the hole and screw to the existing 2x4s. This seems like a better approach than screwing pieces of wood to the top and bottom and mounting the repair to the existing drywall. The problem is that the only way I can think to get that inch is to kind of stab at the drywall and pick away little bits at a time.
Due to all the interest, here's my actual wall/hole. On the left I ended up cutting it past the stud because there was another dent in the wall. The story is that the previous owners, in their 80s, hit the wall leaving big caved in portions of the drywall.