The general rule is that the hole depth should be about one third of the total pole length. In my case, I have a 10.5 foot long pole so I dug a hole that was about 3.5 feet, or a little more, deep. I don't remember exactly what the width should be, but I think with a 4" OD pipe, you're probably looking at a width of 18 or 20 inches. It's also a good idea to put a piece or two of rebar through the bottom of the pipe to make it more torque resistant. You'll also probably want to fill the pipe with concrete when you're filling the hole.
FYI, digging the hole and setting the mast was by far the hardest work involved with installing my dish. Just remember to keep that pipe nice and plumb.
Mike
P.S. Just reread your original post. I'm thinking you lost a lot more than 10 inches when you cut away the post. There's probably a few feet of it still in the ground. It might be worth your while to check out your local scrap yards for a replacement rather than reusing the original.