Yes you can get multiple hat quests consecutively (not concurrently though!) Here's how I was able to do it:
I finished my Yagudo hat quest and it was marked completed in my log. BUT, it's not really completed till you zone back into 'the zone' again. (the zone being whatever zone the beastmen hat quest you're doing is in.) It is not considered properly over until you've gotten your gold beastcoin.
I finished the hat quest prior to the conquest tally (11am est DST for me) but I didn't get the coin until after that.
Once you have your coin it is either a 24 hr wait OR waiting till midnight JST OR waiting for the next Vana'diel day. (I say OR because I'm not sure, two days had actually passed, but since the game doesn't use a 48 hr clock for anything, I'm willing to bet it was one of thsoe). If someone can confirm one of those, it would be wonderful.
After some time has elapsed you may go and get a different hat quest. YOU DO NOT NEED TO TOSS THE HQ HAT FROM THE FIRST QUEST.