When I pick off on threads I always set my sub collet as loose as possible, almost to the point of the part slipping in the sub, or just tight enough to not produce a cutoff nib. (Assuming you dont have any back work to do)
I have had a couple parts that I simply could not grab on the threads. When that happens I zip tie a basket (plastic bowl) to the sub and just let the parts drop in. This might not work for you depending on if you can de-nib the screw by hand.
Final thought: use a non hardened "emergency" or "temporary" collet. I believe both Hardinge and S&M make them. probably wont hold up great but since its soft it might treat your threads a little nicer.