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I think I have the s# correct. The PWDB listing on the Illinois posted by @Waltesefalcon .
https://pocketwatchdatabase.com/search/result/illinois/304981
It is a model 3, fifth pinion model. Basically a transitional model 2 key/stem winder which was designed for use in a hunter case. Converted for use in an open/faced case by adding a ”fifth pinion” to the gear train so the seconds hand would be in the right location. Unusual in that the fifth pinion is jewelled, in spite of the fact it is only an idler pinion.
Close enough for government work, my serial is 304961.
I think decision to jewel the 5th pinion instead of the center wheel a rather odd one.
To the best of my knowledge, the idea of the fifth pinion to allow a hunter cased movement to be fitted to an open faced face, was unique to Illinois. I am not aware of any other U S watch company to go this route.