It may also be worth extending the scope of your request, as "this job or nothing" often gets nothing. Whereas, "this is my trade of choice but I would consider..... as an interim", may get you an offer which is at least a foot in the door.
The other option is to look at the Navy and Army to see if they can satisfy your needs. 3 months is a short time for reapplying, and the scenarios already laid out above are far more common. It sounds like you may have simply dipped it on the numbers game on this occasion; it happens.
Good Luck with whatever your future holds.