Looking to affix medals in a display case
A very simple solution is to use small nails. Your medals slide off the pins because their needles taper off at the end, creating a tiny slope for the medal to slide down.

If you use nails instead (heads facing the front), the heads form a tiny barrier that keeps the medal in place. There's a surprisingly wide variety of nails available. For this purpose you should choose ones with a distinct, but not too broad head and - if available - black nails. 

You might have to add a thin wooden board to the backside of your showcase to give the nails enough material to dig into. If you want to paint the nails in the same color as the felt, do so after hammering them in.