I'd suggest measuring the temperature of the outside water, not the milk itself. If you measure the milk, the water will keep heating until the milk is at temperature, but due to the fact that heat transfers slowly to the milk, there's real issue of overshooting the correct temperature. If you measure the temperature of the water bath, while temperature rises will be slow, the milk will eventually heat to the correct temperature assuming that you stir the milk. As mentioned, measure the temperature of the milk with a separate thermometer.
Adding a fish tank bubbler into the urn water bath will ensure that the water is constantly circulating, and you don't get stratification of heat layers throughout the water bath.
Finally, if you swap the STC-1000 out for a PID, you might get a more accurate system, allowing you to measure the temperature of the milk directly. From my understanding, a PID will 'learn' your system and prevent overshoot/undershoot. That's assuming that you heat the same volume of milk each time. I should clarify - I don't use a PID, that's just my understanding of how they work.