The one exception to that rule is unlock quests. Your new character is an arcanist if i recall correctly. Doing the level 1 quests in Ul Dah(and getting the reward of 100 exp and 50 gil or whatever the payout is on those quests) would be pointless, but doing the level 15 quest to unlock the Gold Saucer would offer your character something new to do and possibly be quite fun.