Adjusting the gain in Pre-Built Circuit
It looks like the answer was indeed R6/R7. These two resistors with the equalization network of R9 - R12 &amp; C7 - C11 do indeed determine the gain.

As suggested, I simulated the circuit, and the default gain at 1kHz was 35db. Halving R6/R7 have raised the gain to 43db. Instead I chose a 1.3kOhm resistor in parallel via switch, giving about 600 Ohm and yielding an output of ~40db.

I also checked the RIAA equalization in the simulation for all three scenarios, and it's true that the equalization is slightly affected, the low-frequencies which are slightly more attenuated (less amplified) if the gain is increased. For 390 Ohms the deviation at the 20Hz end is about -3db. The equalization of frequencies from 1kHz and up isn't affected for some reason.

I picked 600 Ohm as final value since the deviation at 20Hz is about -1db VS the ideal RIAA-equalization curve and this is probably inaudible, especially since my speakers top out at around 40Hz anyway.

Upon implementing the above discussed changes (with a toggle switch) it seemed to bear the simulation results out.