Perhaps it did migrate, but someone initially stated that 4-jaws would be better, without referencing hobby machining.
I think there is a reason why self-centring 3-jaw chucks are the usual choice for watchmaking. I rarely do any eccentric turning, but I do have a faceplate which I can use.
Back to the original post: I can recommend getting a Burnerd or similar quality 3-jaw (German, Swiss, English, American). The Sherline chuck which jdrichard describes sounds promising too.
Further to John's remarks,
-I would advance the hands by 12hrs, if the fault is in the calender mech then it should loose time during the day.
-Note how long it runs on bench following a full day on wrist.
This points to the area the fault is at.
Does your app measure amplitude?
Then what was your definition of fixing the watch in other words what have you done to it?
One of the reasons automatic watches can have issues is if they just run out of power. It's why asked if your timing app has amplitude measurement. So in other words when the watches on the wrist it manages to keep enough power to be running as soon as you take it off your wrist did just runs out of power. Then next day's your watch still running or is it just off by a couple hours?
Another Guess since basically were guessing without enough diagnostics information. I see from the picture at the link below it appears to have a calendar mechanism? There is possible you're losing your time during the calendar change but as I said it's just a guess.