I have a somewhat similar system, tube preamp, Meitner MTR 101s recently updated by John Wright and a Bluesound node 2i. I am using the DAC within the 2i, but what made a nice and noticeable difference was going from the ethernet cable that came with the unit to a Supra CAT8+ purchased from zendada on Ebay for about $50 including shipping. He calls it CAT8+ but it really is CAT8. Not sure how much, if any difference it would be going to it from your CAT6.
If you do go this route be sure to purchase it from zendada, the other ebay seller is in England so shipping is another $20 and takes three or four weeks to get here.
Besides the Supra CAT8 cable if you chose to, I wonder if you need new tubes for your preamp instead of a tube DAC. The MTR 101s are somewhat "tube like" and you already have your tube preamp. I only mention this as I had a substantial and surprising improvement when I changed my preamp tubes, the old ones lose their sparkle so slowly you don't notice it until you change them.
If you do go this route be sure to purchase it from zendada, the other ebay seller is in England so shipping is another $20 and takes three or four weeks to get here.
Besides the Supra CAT8 cable if you chose to, I wonder if you need new tubes for your preamp instead of a tube DAC. The MTR 101s are somewhat "tube like" and you already have your tube preamp. I only mention this as I had a substantial and surprising improvement when I changed my preamp tubes, the old ones lose their sparkle so slowly you don't notice it until you change them.