It appears that if you book your ticket online you pay half price. There are some constraints but you can use your credit card to pick up tickets at the station so, in theory, you can book on the day of travel (I think you have to allow 2 hours for Southern's web site to process the order). You can book further in advance and they will post you the tickets to you; this is separate from the 40% off advance fares deal.
http://www.southernrailway.com/offers/50-off/.