The checker is not showing how much distortion there is. I recommend using a preview texture with a finer grid, and circles, and letters, and a fine per-pixel grid... for example: http://wiki.polycount.com/wiki/TextureCoordinates#UV_Map_Grids This is my current fav:
You dont really need to do the split on the top side. Just outline the circumference of the top side and of the bottom side with seams, and unwrap that. I havent looked too much into the new Peel tools so I dont know if this was done by AutoPeel, but it looks like that's what happened here. Some manual placement may be…