Forgive me if I'm stating the obvious, but your chain tool **does** have dual sets of chain holder pins. The "far" set from the punch is for driving pins out/in on the chain, and the "middle" set of teeth is for you to put the chain in and then gently tighten the punch to push the far chain link-wall out just slightly. If done properly, the chain comes out flexible each and every time, and you don't have to manually flex the chain to get the link loose.

You probably know this and I've just insulted you by implying that you don't... No insult intended, but the chain tool gets my links "like new" each and every time (I use the less-expensive of the Park tools).