How deep are standard female pin headers?
the ones I have here are 10.5mm board-to-board when mated to a standard pin (pin 8mm tall above board) so 8.1mm for the header body and 2.4mm for the pin strip.

this can be stretched to about 13mm before they let go. so for a 15mm gap you want a header than finishes between 10 and 12.5 mm above the board.  

or you could use and ordinary male header and use a female header with long pins.