Haha, my response was a bit tongue in cheek.
Advertising will help but the one thing they lack is capturing consumer who searches directly with airlines. I'm sure many on here (and I know many people) who start their search directly with an airline and if they're reasonably happy with the time and price they just book it.
Obviously, there are other ways they can capture booking (travel agents, BFOD policies etc). Rex somehow needs reach these people by jumping up and down saying they're better/cheaper than the competition.
As someone mentioned up thread, flying people for free (or making them just pay taxes ~$25) would be a great start. An alternative would be to quickly launch a frequent flyer program and make people want to use it (status matches, bonus points/status, fly 2 flights and 3rd is free etc).
One of the most genius pieces of marketing I've seen this week was BWS. 1hr guaranteed delivery or your next delivery fee is free. Chances are they'll make the delivery time but if not they bring you back to make a new purchase (because you'd get free delivery next time). Rex could offer free extra legroom seating or some other perk on next flight if the flight is delayed (this sort of thing captures media attention).