The daily expert roulette is probably the fastest way to get a map, unless you're unlucky enough for 3 or more S ranks to spawn while you're in the dungeon. Since it's just once a day though it's a bit lacking. I like to do the trial roulette because it's 110 or so myth in about 5 minutes (usually).

I hate hunting, but it is unfortunately (or fortunately) very fast/ practically without restriction. Some of my friends zoomed way ahead of me on alexandrite acquisition by taking a few days to hunt for several hours. Seems like any time I decide to hunt though there's a mysterious 45 minute+ downtime where absolutely nothing spawns...