Basic principle of noise versus gain:
Put a gain of 10 in front and use 6V instead of 0.6V (forward drop) diodes. Now the noise (due to the distortion stage) is 10 times lower. Attenuate output by 10x to fix output level.
Of course, you don't have the dynamic range on a 9V to do that (+/4.5V swing, and that's with an R2R amp), but that's the right idea. You could use 3V zeners, though they're rather soft (probably soft enough to be okay, considering Vf is even softer). Or stacks of regular diodes. Or blue LEDs (~3V Vf -- bonus points for putting them on the top cover as a loudness indicator?).
I have to wonder if it really makes so much difference having 100 gain at the low level; why not 10 or 30? Or if it's the clipping that matters, ramp up the gain so it's got a hysteresis band instead; now gain around zero input is completely zero, and slightly larger signals are heavily clipped as usual.
Not really much you can screw with, since, as far as I know, guitarists are extremely picky about their "sound". You can scale things linearly up and down and it won't make much difference, but if you start playing with the range and break points of the nonlinearities, you'll "change the sound".
As for just... low noise op-amps, maybe an LM833 would have better results? Not much difference in the basic noise level of most general parts, and I expect it will be easily swamped by ambient noise, EMI and stuff like that.
Tim