The copper cable makes a very small and subtle improvement, so the first step is to optimize your source and amp/tubes. Prior to getting my HE-500 and HE-6 I would most often use the HE-5 with the HE-5 LE cable, and I used the HE-5 LE with the slightly brighter original HE-5 SPC cable. I discussed the increased treble and air in the HE-5 LE with the SPC cable in my HE-5 LE review. I did find the copper HE-5 LE cable to smooth out the highs of the HE-5, but at the expense of a tiny loss in micro-detail. So, I'd guess that the HE-6 cable might be the best choice for it's extra micro-detail.
I have not tried any other brand cable with the HiFiMan phones, and I'm still at a loss to explain why a cable makes a difference in the first place. But I clearly recall how I couldn't believe it the first time I used a headphone extension cable made from Grado cable and I heard it kill the highs and micro-detail in my HD600 with APC V3 cable. Blutarsky was there with me, and we both heard the improvement when we removed the Grado extension cable from the signal path.