Some possibilities: (I will try not to repeat the good suggestions that the others have already given)
1. Is the cable you are using to connect the HP-1 to your Aux Input of your car a stereo cable?
Many scanner speaker outputs are Mono but the headphone and Line-Out are usually stereo.
Yes, it is a stereo cable. I also tried to use a mono cable with a mono to stereo adapter in order to get the sound to come out of all four speakers, with the same result.
2. Did you try the "Line Out" from your HP-1 to your Aux Input, just to see if it would work?
I could not try this initially because the line-out on the HP-1 is a 2.5mm plug, while the Aux input in the car is a 3.5mm plug. I also did not think it would work because I assumed that the line-out does not provide any power, so it could not drive the speakers. However, I had nothing to lose, and since I tried everything else, I went ahead and bought a 2.5mm to 3.5mm stereo adapter. I then used it to connect the line-out from the HP-1 to the same Aux port in the car. It worked! I tried this without the ground loop isolator, and there was no hissing. The audio was loud and clear (as clear as can be on a digital channel). I can't say that I have an audio issue now, since the sound is as clear as that from the HP-1 speaker. And as I said, I no longer need the ground loop isolator which I was using with the headphone out to Aux in.
3. When you connected your "Headphone Out" of your HP-1 did you try adjusting the volume of the HP-1?
I tried adjusting the volume high and low, with no improvement.
4. Have you checked your car's owner's manual for any adjustments or approved uses for your Aux Input?
I did, and there were no instructions regarding the Aux input.
5. What inputs were recommend for your FM transmitter (speaker out, headphone out, line out)?
If your car has an extendable antenna did you extend it?
I stopped using the FM transmitter since the ground loop isolator greatly improved the sound quality.
6. Distortion & hiss can be a sign of overload (assuming your antenna, coaxial cable & connectors are still good). Radio Shack does sell a audio "Attenuation cable" but all the ones I have seen are mono so you will need two mono to stereo adapters to use it.
I don't think this is the problem. It looks like most of the issue was caused by a ground loop.
However, since you can turn the volume up and down I would think that should have took care of this problem, so I would hold off on the "attenuation cable" as having a low probability of helping at this point.
The suggestion to have you HP-1 on battery power and not hooked to the car power (just to try it out to see if it helps) is a good suggestion.