Like @Monsterzero said get an OTL tube amp like a Bottlehead Crack or Schiit Valhalla (great pairing with 6XX) to go with your 6XX and enjoy. I'm saying this as an owner of the 58X, 6XX, 600, and 660s.
An OTL tube amp will truly bring out the best of any of the HD600 family to the point you may not feel the need to upgrade after that. If you do feel the need to upgrade in the future as budget allows I also recommend looking at several models of ZMFs (since you like the HD600 family type sound). The OTL tube amp will be a great match for those also. Just my dos centavos as a fellow enthusiast of the HD6XX/650.
An OTL tube amp will truly bring out the best of any of the HD600 family to the point you may not feel the need to upgrade after that. If you do feel the need to upgrade in the future as budget allows I also recommend looking at several models of ZMFs (since you like the HD600 family type sound). The OTL tube amp will be a great match for those also. Just my dos centavos as a fellow enthusiast of the HD6XX/650.