I have a set of Kimony awls, I use the setting off often. I just work each cross, poking and pushing the cross until it's straight. It's easier when it's still mounted, but you can still use it unmounted. I also have one that came from a Prince tool set. The handle bigger not narrow like the Kimony.
The setting off tool is the third from the bottom.
The red handled tool in the Prince set.
The one that looks like a mini meat hook, is not really a setting off tool, it's a string mover/puller mainly used by badminton stringers. It's great for moving the string in a shared hole. I've seen a great video of it being used instead of a pathfinder awl.