I’ve never heard someone refer to a European dish as having umami, it’s just upper class people showing how cultured they are by knowing the Japanese word for savory flavor. To top it off it only gets used when talking about the big three Asian countries. No one is talking about the umami profile of Vietnamese food. It’s the same kind of Anglo person who would read Sun Tzu while drinking lapsang souchong in the mid 20th century. Both are fine things, but together you know this guy is sad about the fall of the British Empire.
I don't add anything to it but I also seem to like smoky flavors more then the average person. I don't think sugar would work very well but milk might be interesting. You can also try adding a small amount of it to other green or black tea, not enough to overwhelm the flavor just enough to give it a smoky backbone.