Thanks for the info, Lordgalvar and Friendship. I will probably pick up a noise gate at some point and plan to inquire about an internal trimmer.
The Gonkulator oscillation noise is more audible with higher gain and frequencies, though still present at lower frequencies and with the gain knob at 0. I use isolated DC power and quality cables, am currently powering the Gonkulator with a battery, and experienced the same problem with the pedal isolated. The noise is faint for the most part, but stil annoying.