What works well with our machines:
Overdrive, fuzz, and distortion pedals often have a less noticeable vibe at the output in my tests but your mileage may vary. I have had some success using a fuzz to square up a weak signal; a current project that involves mic input is calling me back to the bread board!
One of my favorite pedals is delay. This type of effect really open up the possibilites in how you "play" your lunetta. When looking for one, look for long delay times or at least one that can be modded to give longer repeats.
My second choice is my trusty MXR Phase 90. The best way I can describe it: chewy! Seriously, this effect adds a ton of character to these square waves.
Thirdly, a wah wah pedal is essentially a bandpass filter that's adjustable with your foot of course. Sweeping from low to high easily allows for more expression during jam sessions. Extra bonus points: we don't have to dig into a lot of theory to achieve filtering effects. ( We'll discuss some simple voltage controlled filter concepts in the near future though).
With wah wah pedal: