Problem solved.
Adding the MV potentiometer required me to run a long cable between the slider of that MV potentiometer and the grid stopper resistor of V2. I had opted for a shielded cable. But I hadn't connected the shield to ground.
I've just soldered it to the terminal on the potentiometer that's connected to ground, and the noise is gone.
This kit from Tube Town is truly excellent, well-made, and has a beautiful tone. I highly recommend it.