There's a little part called a detent that acts as a lever when the stem is pulled. One part of this part "snaps" against a spring so as to stay out when pulled. It is actuated by a pin at one end that sits in front of a shoulder on the stem. When the stem pulls out it pushes the pin and moves everything to setting mode. This pin also hold the stem in.
But on this watch the plates themselves are so worn that the stem can move around too much and get out from behind the pin, and thus come all the way out of the watch.
Secondly, I filed away a minuscule portion of the lever end so that slightly less pull is needed to snap it into setting mode.
This is a 20/0 size watch; almost as small as they get. I should note that one of these photos above shows two of these parts. One in in place in the movement and the other loose. I tried replacing the detent first, but no luck there fixing the trouble.
Interestingly, instead of dial foot screws, this watch has two screws, at the top and bottom respectively, holding the dial on.