I've found doing a combination of Dungeon Roulette, Hunting Logs, and Leves to be most effective. I've never had much time for FATES, and only do them if they're near me and I find it convenient to stop and participate.

And if at any point that gets too monotonous, I'll swap to BTN, gather and make some gil for an hour or so, then swap back and keep right back at it.