Just to bump my own thread here: I spent a bit of time putting my OD2 in front of the Jet City. I had the pedal in stock Tube Screamer tone (aka mid hump and default TS clipping diodes), level at noon, gain a little less than 9:00, and tone rolled all the way off. Yes, you read that last part right. Probably would sound like crap with a regular TS, but my OD2 is brighter by its very nature and the Jet City is a bright amph. FWIW, the amph was set with the gain, bass, and mids at 6, the treble at 3, and the volume and presence at 4.
The difference was subtle but definitely there. Not sure how to describe it . . . more saturated maybe? But it didn't lose any clarity. It's kind of like adding salt to a dish. You don't want people to taste salt, you just want to kick the existing flavors up a notch. Mission accomplished. It gave some nice bluesy tones when I moved the amph to some clean settings (gain about 1 and volume to 5 or 6) and switched to the neck pickup.
Mind you, I have yet to mess with the Full or Fat tone switch settings, to say nothing of the diode lift or MOSFET clipping. There's more to be played with.
I'm still interested to hear how others use an overdrive, especially with an already overdriven amph.