If I recall, whenever I've had a ticketing agent add me to a stand-by list, they print out a "priority verification" boarding card, which is basically like a boarding pass for the standy-by flight except it doesn't have a confirmed seat assignment. I've used those to get through security before - may work for your situation?
(Or others may tell you that using your 6am next-day boarding pass will be just fine, in which case ignore what I just said
)