For manual finding the optimum current, one could look an the stabilization curve (the first 10-20 seconds or so) after turning the current on. For too low a current, the voltage should drift down on settling and for to high a current the settling should be upwards.
Of cause one can do the same with the SMU and PC as well.
As heater one could use the second zener as well, so the observed voltage will stay about the same and one could use an AC coupled amplifier to watch small changes.
The much better performance at 10 mA suggests that the wandering / drift was due to thermal effects and 10 mA being much closer to the point of zero TC. Only 6 µV suggests this is already rather close - e.g. a TC of a few ppm / K at most. Final performance can also depend slightly on the resistor to set the current and the environment temperature range.
Short answer: the voltage never settles upwards.
The datasheet that Zlymex posted has max power at 200mW and current of 30mA.
I have tested one sample at 5mA, 10mA, 15mA, 20mA, 25mA and 30mA. The device is wrapped in three layers of tissue to kill any air current.
The voltage always decreases for many minutes until a stable voltage is reached.
The particular device in-hand operates at 6.19xxxx volts at 30 mA. I will let this device run at 30mA until it settles and report +/- deviation.