i use tw leather ones (under a babolat originals) and the leather grips do seem to settle down permanently, just after a few hours of play. now the change is insignificant when i replace my overgrips. i didnt stretch the leather grips before putting it on and i still dont get any problems.
i get the same frustration as you because everytime i put on a fresh main grip, it definitely feels much more thicker and i have to adjust to it. so this solution definitely worked for me! yay!
babolat apdc, SP Tornado, 17G, 55lbs
last string :technifibre bi-phase x-1, 17G, 57lbs