You just need to do more quests if the moogle doesn't offer you the quest... In my opinion, if you can get the goblin bag expansion quest and the RSE quests, you should be able to get the mog expansion quest too. Just talk to the moogle from time to time.
If it still doesn't offer you the quest, try talking to the moogle at home country and also in rental house. I got the quest offer on a Sunday (real time), but I don't think that makes any difference.
As for Archaon's question, I don't think the mog ever "leaves". Just wait several VD days and talk to the moogle again. By then you should get your 10 extra slots
