How to cut out precise inside corners from hardboard?
A couple ideas:

Try different jigsaw blades. Many different tooth configurations exist, and some will work better with fibrous material.
Use a circular saw set to full depth. The cut angle will be nearly 90°. Clamp guides to your workpiece if you don't feel steady. You can finish with a utility knife once you snap the remaining tip of the scrap, or overcut slightly on one face if that's not a problem.
Use a rotary tool and a jig.
Use your jigsaw, then clean up with a rotary tool and a straightedge (or just a utility knife).
Finish the cut with a very sharp chisel. The flat (non-beveled) side should be against your final piece.