Pix....

Excellent to send the file. The issue I think you are having is the amount of influences you used when you bound the skin. Try unbinding the skin, then smooth binding it again with less influences. I tried you rig with just 1 influence and the chest stopped moving with the jaw without any other touch up.

Such a low influence might cause you other problems when the whole character is moving around. You're just gonna have to play around.

Also, just a question, why did you use a skeleton for facial movement. I have heard that the best way to accomplish this is with blend shapes. Blend shapes is more complicated and your character is pretty simple. I was just curious why.

Otherwise cool rig.