Like Steve, I hollow late in the process (like just before I glue and final sand).
I use a drill press and 1" or 1 1/8" forstner bit. Do not scrimp on the bit. I personally like the ones with the serated edge, especially for tipping and cutting at an angle.
For padding, I use several layers of 1/8" or 1/4" foam core board duct taped together in a stagger/stairstep pattern. Putting this pad under the top, rounded part of the deoy allows me to drill the top piece at an angle as well as perpendicular, by rolling the top piece up on an edge. The foam core prevents damage to the edge and holds the piece well. When the boards get worn out/compressed too tightly, I just make some new ones, but a pad usually lasts through a dozen or more hollowings.
For final thinning, I use a carbide cutter in a foredom handpiece to get right up to a super hollow decoy if that is what I am going for. As a final check after hollowing, I screw the body together and check its balance, side to side and fore to aft, balancing it on a steel edge on the two centerlines. If the head is offset or a radical turn, I usually screw the head on too when balancing. If I want the tail down, like a diver, I make sure I keep the front light and leave a little extra in the tail area (as the head adds weight that can push down on the front and lift the tail). When you check the fore aft balance it will show as slightly tail heavy at the mid point.
If I want the tail up, like on a woodie or pintail, I will make sure I make the tail area is as thin as I can and leave a little extra on the breast/head area. In this case, the decoy would be slightly front heavy. If you have a tail or other area that needs to be hollowed but cannot be easily reached by a carbide cutter, a 1/4" high speed drill bit, inserted in a foredom handpiece can, with care, be used to drill pilot holes and then be used to egg out those holes by cutting with the drill edge
If you want, you can go nuts on this idea of compensating with hollowing versus non-hollowing to overcome radical head positions, offset heads, etc. Or, you can just make it as hollow as you can, and then use supplementary weights and a keel to make it ride the way you want.