Your chuck has 1 pc jaws, hence a set for each ID & OD. Generally soft jaws are made for 2 pc reversible jaws (tongue & groove).
You could make a set of soft jaws from scratch but that requires making the teeth for the scroll as well. For the amount of work required it's not worth it or practical as it will wear out fast if you make 1 pc soft jaws.
I'm not sure what ops you need to do but in your case making an aluminum (or any material you have on hand) face plate might suffice (either to mount in the chuck or directly on the spindle). You can then use use epoxy, CA (super glue), or double sided tape to attach the disc to the face plate. Using heat will allow you to detach the part from the face plate easily.
I've alwaysed used epoxy rather than CA though. Takes longer to set but the epoxy is easier to clean up with heat. CA is quick to set but I hate the smell of CA when heated.