Law faking issue

I am trying to use a 100k lin pot with law faking resistor for volume control. Initially I put a 47k resistor between the taper and GND, but then the volume was too high throughout most of the range, and the change at the bottom end was very abrupt. Then I replaced the resistor to 6.8k. Now the volume is acceptable, but the volume difference between say 1/5 and 4/5 (30 degree to 270 degree over a 300-degree range) turn is quite small, about 20 dB. Normally I never use it over 4/5 (which is 20 dB below maximum BTW), but I want to use it below 40 dB attenuation, which is below 1/5. My problem is that resolution and channel tracking is not very good there. It seems to me that law faking with good resolution is usable over a 20 dB range only. Is there any trick to improve it?
setting the faking resistor to between 10% and 15% of the pot value.
Try some values between 10k and 15k.

The LIN pot is recommended only because of better tracking at low volumes.

But your Zin >=5times Rfake and preferably >=100k.

You may find that Zin is already working as a law faking resistor if it is below 20k.