the thin part of the flat prob came tender and went tough without you catching it.
i wouldn't separate.
use the thin part for burnt ends and be all good.
cold meat=good smoke ring
once your coals are nice and hot, throw on wood, throw on meat, right out of fridge or cooler. don't worry about white smoke with the wood, it'll turn thin blue pretty quick and not affect your flavor.
DEFINATELY do not wait until the thin part gets tender when the thick part already is.
how much wood do you use?