Put a bit of celery salt in your rub & use nothing but oak wood for smoke......
Chill the bugger good & go straight from the fridge to the pre-heated pit, no piddling with sitting on the counter or 'coming up to room temp'............
EDIT:
This was kinda dumb of me.......the REASON to go on as cold as possible is that smoke ring forms only up to 140f......in other words, the 1/8"~3/8" inch or whatever you get will happen near the surface only under that temp as it's heating up at the start.........
Another possible help might be to start the cooker out at fairly low, creeper heat (not always the best idea to play catch-up, but if you have a good handle on doing that, it will keep the temp on the meat surface lower, longer)
Stuff will take on smoke as long as it's applied, smoke ring, only up to 140f