How is this tool called?
Attaching picture for reference.
The hat making trade call that a ''Pusher down'' tool. It is a very old tool used mostly in the making of men's hats but does have the same use in millinery.
It is used to push down a cord or thin rope when blocking crown blocks. The rope has to be quite tight and this tool enables the pushing down, thus saving fingers from hot steam or water. It helps keep the break between brim and crown in place. Other tools equally as important to having one of these is the ''foot tolliker'' that is used for putting in a sharp crease at the base of the crown and the beginning of the brim, called 'the break'
There are many versions of this tool over the last 150 years as each maker of them has there own style. Some have a cord groove in the tool, other not. It can come down to preference.
Hat Blocks Australia make a version of this tool.
I hope this is of help to you