A properly cut and lubed nut, and as few string winds as possible will solve most tuning roblems on any guitar.
once I started stringing like this,
My guitars have super stable tuning, I know I've said this several time before, but its worth repeating, all the extra winds on the posts are a major culprit for tuning stability, a ton of winds will never get all the stretch out, few winds and a natural lock as shown has worked wonders for me, plus its a lot quicker and easier to string up. The pic is from my Gretsch 5120, I had all kinds of trouble staying in tune using the trem till I switched to thos method, now I can yank and slam the arm as hard and as often as I want and tuning is solid, I use this methos on all my guitars except for my slotted post vintage strat tuners , but still use very few winds on them as well, you just lock the string differently.