Looking for a mechanism similiar to a watch's stem in rotary encoder type form
Yes - many oscilloscopes and other pieces of test equipment have rotary encoders with push/pull switches on them.

An example of one like this.

Note that I think some rotary encoder classify the outputs of the rotary encoder themselves as push/pull, so this may confuse searching.

As stevenh comments, you may want three levels - free spinning, time at first level and then date. I have one seen a potentiometer on an piece of sound equipment with this functionality, although the neutral position was in the middle, with the in/out being spring return.