I don't believe the coldness of the brisket has anything to do with the smoke ring. I let my briskets sit for about a half hour to 45 min and I get great smoke rings.
It probably has more to do with how much smoke your producing. I believe there's a thread about the salt content in the rub. Never brined either, so can't comment on whether that effects it.
Diamond Plate FAT 50 Ultra
55-Gal Horizontal offset
2 55-Gal Vertical Barrels
Anything worth fighting for is worth fighting for dirty!