You can't do chocobo races while waiting for a queue. You cannot do any instances while waiting for an instance, so no instanced quests, no lord of vermilion, clearly no runs with your squadron etc.

However, you can always go and play the card game, hunting for new cards, if you're interested. Or do the side quests as SenorPatty said. Some of them offer decent experience, and even if not, they are something to do that progresses you somehow.

Alternatively, you can find a tank/healer manually (not necessarily a friend, really, just can try and use the chat) and then queue with them in the party. This will not be instant-queue, but hey. It's still considerably faster with one of these classes in party. Doing a party find for x consecutive runs of a dungeon will be decent option if you've got that extra time to play and want to make use of it. Once you do find the party which may take a while, you should be able to get three or so runs at least with minimal wait time.