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 2×4. 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.
Edit
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.
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Best Answer
You want to add backing to the inside sides of the existing studs as well as to the centre of the span. The simplest way to do this is to cut the new backing (preferably 2x4, or whatever the existing stud dimensions are) about 4" longer than the height of the opening and screw them to the existing studs. The cut one more piece of backing to the same length as the other two, but this time place it in the centre of the opening and screw through the top and bottom of the existing drywall to attach it. See the picture below for reference.
This method will be much easier than trying to score the existing drywall back on the existing studs and will provide a solid patch.