They don't necessarily have to have a lot of string follow. String follow can be kept to a minimum if you spend some time to get the belly taper just right when you rough it out, and make sure the tiller is spot on before you pull it past brace height.I'm still trying to get that one down myself, but I have done it at least once.
BTW If you want one to have zero set it's prolly best to introduce around 3" of reflex first. You didn't mention how much set you had but,between one and two inches of set is pretty good for red oak, and even three is acceptable.
One caveat though, wood is gonna vary in density from board to board, so you can do everything right and still end up with more string follow than you wanted simply because the particular board you used was not as dense as you would have liked.