I usually put them on to wax paper because the paper towel is too absorbent. At this point, just roll it on to the wax paper, then turn it so it won't get lopsided. Sounds like it doesn't have hooks at the end of the cremaster any more; otherwise, I would have suggested tying some dental floss to the end to hang it up.
You can also take a cotton ball and use a bit of cotton, but you need the hooks to do that. You can then stick the stick pin through the cotton. I'm afraid you can't stick a stick pin through the cremaster. It is like a hard stick.
They can emerge from a laying down position, but do need some sturdy to crawl across. After it has dried, you can put it back on the paper towel. I use net containers so that they can climb up to dry their wings.
Herndon, VA (USA)
Take care of the small things....