Sounds like thermal noise. When you roll the volume off on your guitar you increase the resistance of the source (your guitar) greatly. 70% is about the worst possible position in this regard for a typical audio taper pot.
For example, assume your guitar has a 500K volume pot with an audio taper. At 70% you're about halfway on a typical audio taper pot so the resistance "looking into the guitar" is about 125K (250K in parallel with 250K). That's about 12 dB more thermal noise than if the volume pot was wide open (assuming a 10K pickup resistance).
Furthermore you've reduced your signal by 6 dB so there's a net degradation in SNR of 18 dB, which is huge.
So the answer to your question is "No, you are hearing the noise of your guitar. You can't beat the physics unless you want to reduce the temperature by 100 degrees".