A low output Moving Coil (LOMC) cartridge needs, in addition to the RIAA equalization, 20 to 30 times as much gain as a Moving Magnet (MM) cartridge.
To get low current noise, a FET input stage is needed. To get low voltage noise, a BJT (ordinary transistor) input stage is needed.
Given that LOMC and MM cartridges need different input stage technology, you can't just add more gain to your MM phono stage and call it a day.
The output from a LOMC cartridge is so miniscule that just adding more gain would be detrimental to the sound quality. (Yet, this is what most DIY projects do, and surprisingly many commercial offerings as well...)
To get excellent sound quality from your LOMC cartridge, use the Muffsy MC Head Amp in front of your MM phono stage.:
If you have the Muffsy Phono Preamp and MC Head Amp built in the recommended enclosure, or you're using another MM phono stage, you can connect them like this:
You can build the Muffsy Phono Preamp and the MC Head amp into the same enclosure (only one Muffsy Power Supply is needed for this), and switch between them. This requires your build to be in a larger enclosure, so you won't be able to use the Muffsy Back Panel.
You'll need two sets of RCA inputs, and a ground screw, to switch between MM and MC. Use a 3PDT (three pole, dual throw) switch to select between the two inputs, and one set of RCA outputs.