I use 4-5oz veg-tanned and whatever-dyed leather soaked in warm water for ten minutes. The soaked leather will stretch and dry tight, which is why you leave the gap when you trim it to size. Punch your lacing, I do four holes to an inch most of the time and lace it good and tight with fine twine, leather cord, rawhide cord, shoelaces, etc.
I read
this trick that Wrap the handle in masking tape the wrong side out and cut straight down the belly/archer side of the handle to transfer the rough pattern, then put it on the handle and trim it little by little until there's maybe an eighth-inch gap between the leather.