Here's what it says on the FAQ about Ex Level.

"Ex Level
If you clear a One-Shot Mission's corresponding mission, your Ex Level limit will increase.
After an Ex Level limit is unlocked, you can raise all your cards to the upper limit."

Safe to say in this version of the game we've not encountered any of those missions yet. I got an Ex Card as part of my reward from the event (rank 25 whoo!), finished story mode in its current state, turned in the quests and I can't use my Ex Card.