There isnt really a "set" time that you start getting better gear, there is an event around 60 hours or so that has a higher chance of getting a rare piece but ive never noticed much around that point. Other than that its mostly random, Ive had dwellers out for 3 days bring nothing but garbage back, and dwellers out for 20 minutes grab a legendary.
luck stats raises the ammount of caps/loot you get, but doesnt seem to have much effect on loot quality, it honestly feels like the system favors quantity of explorers of quality, so just keep sending them out and hope you get lucky.
I have noticed as a vault grows, the quality of (at least common) weapons improves, when you first start you will get mostly 10mm and .32 pistols, then move up to lever action and hunting rifles, then sawed off shotguns and laser pistols and so on.
TL;DR - No real sweet spot, mostly random, just keep trying.