The sounds off the turntable needle directly contains a lot of scratchy, high-frequency noise. The output is also too strong for the LM386 amplifier directly. After some tinkering with low-pass filters, I found some appropriate values get a good sound. I brought out the bass a little more than the original record player, and at this point I find the sound more pleasing than original amplifier.
I will publish the circuit schematic along with the PCB.