I would like advice on how to drill or bore a 7/8" hole in the butt of a shotgun stock. It already has a hole in it that is off center of the hole I wish to bore. I need to bore the hole so that it is parallel to the top of the stock. This will accommodate a recoil device. The recoil device is cylindrical and absorbs energy to give a less perceptive recoil. I have a drill press, lathe and milling machine. Holding the stock is a real challenge in itself. Finding a mill that will bore a 4 inch hole is another concern. My mill will not hold larger that 3/4". I am interested in learning types of bits and rotational speeds that would work the best. This is all new to me. The wood is walnut.
You apparently are sensitive to the high likely-hood of the the drill bit wandering off course when attempting to drill a hole that partially overlaps an existing hole.
The usual approach when dealing with this problem is to fill the existing hole. Fill it with a snug fitting dowel, ideally of the same type of wood (in your case walnut) so that it will react similarly when the drill is applied. If a very soft wood were to be used there could still be significant wandering.
Once the old hole is filled, the problem is no different from drilling into fresh wood since the tendency for the bit wander is eliminated..
As for which bit to use, I would most likely use a Forstner bit, but you could certainly get away with using a twist bit, or if you're really gutsy you could go for a brace and an auger bit :).
As for securing the work piece and lining up the drill path, that's an entirely new problem, but you should be prepared to test any setup with expendable wood before you attack your precious stock.
I Can't say I have done this much but here is what I found to work the best for what you are describing.
I would recommend a forstner bit. This bit has a point in the center that helps with keeping it in the center. This is the bit I have used and had the best luck.
Other drill bits that I use I would stay away from. Spiral drill bits will wonder off when only half the bit is on the hole. Image below to help understand what I mean. And A spade bit is also not a good idea. I have used it in wonder if it would work and it does not. Again it would like to wonder off but this time it cuts in a ugly way. More smacking I guess you could say to the wood.
I can not recommend other bits since I have not used them. Clamp the work piece if you can. Hope this helped a little!