Dan, I have done a lot of reseach on this set of yours.
It seems that the secret to this “PUSH PULL TUNING DIAL” Knob is that it is job of that Fourth knob which is the one located on the front of Cabinet to the far Right of the knob line-up. Its got to be one of those four knobs. I think it has to be that fourth knob which ever one it is. And for some reason it is not mentioned on the Pacific Data Sheet for your Model 10240. This has to be the Push Pull Knob we are looking for. Maybe it is frozen or possibly just spins freely until one pulls out on it???
I have not been able to find any other pictures of your actual chassis on-line. But I was able to find your model on Pacific. See below.
This complete setup as stated by Steve is called “Swing Dial or Swing Tuning”. It was promoted by Sparton as “SELECTRONNE TUNING” And yes it does have a dial string mechasim, but it is clutch operated. And maybe, if you have already removed the chassis from its Cabinet, and while pulling off that knob, you may have already released the clutch????