After contemplating all my different design ideas, I decided to start with the basics and make a prototype. This called for a a simple piece of wood shaped like see-saw.
Even with this simple design I suggest a few variations: Cutting the whole shape from one block of wood; Cutting a dowel rod in half and gluing it at the middle of the auxiliary jaw; Inserting a short dowel into the auxiliary jaw to act at a hinge.
I ended up making the adjustable jaw by cutting its shape from one block of poplar. A harder wood like Maple would be even better.
And here is how the jig works...
You can also hold the workpiece upright or at an angle. To increase the jaw's grab you can glue a piece of sand paper on its inner side and maybe on the vaulted hinge back.