I have a few interior doors that do not latch closed because the bolting mechanism and latch hole are vertically misaligned. If I remove the strike plate I can extend the size of the latch hole, but then one of the screw holes of the strike plate ends up over the (now bigger) latch hole and the screw won't hold.
(Note: I don't want to move the door hinges since the doors align nicely in the casings.)
If you can, glue a plug into the latch hole and similarly drill out the existing screw holes with a 1/4" bit and glue plugs of 1/4" dowel into them.
Now you have a blank slate to re-locate the strike plate wherever you want to. Use a spade bit for the latch hole (3/4" or 7/8" or 1" - use the whatever is closest to the size of the strikeplate hole).
The answers above are the correct and proper ways.
However, an alternative way is to temporarily screw a piece of 3/4" wood over the new hole location and drill from there. The plywood will keep your hole saw straight. Then you can purchase an oversized plate made for the thickness of your door that wraps around the strike. Doesn't look good, makes it look like you repaired door when after it got kicked in.