Agree...put the absolute first date you can really make. I got called for an interview in April with a July avail date. They then called me and asked if I could make a June class. Had I tried to game it and put a June date instead of a July avail date, I may have had an uncomfortable conversation with someone. Just look at Delta right now...they were hiring into a 4 month pool, now the pool is empty. You never know.
Talk to your command and sit down and figure out when the first absolute day you can stop working there and start working full time somewhere else. That is your availability date.