If you find the HD600 thin, then barring upstream issues (your current amp is probably good enough to drive the HD600 reasonably), there's a good chance you're in the camp that greatly prefers the HD650. I'm in that camp. They have more & tighter bass, more weight to the presentation, lass treble energy, and are overall smoother with less grain. I find that despite the "darker" signature they're actually MORE resolving than the HD600.
On cables - with the HD650, I found the Zu Mobius cable best - better dynamics, better bass, and returns all the "airiness" that I thought was exclusive to the HD600. With the HD600, the Cardas was a good match - the Cardas does smooth things out a bit, but still I prefer the HD650 w/ stock cord over an HD600 w/ Cardas.
The HD600/HD650 headphones, more so than most even many higher priced headphones, respond to source, and especially to a quality vinyl setup.
On amps - well there's a whole word out there. Best I've heard was a headamp Gilmore Dynamic Reference for SS, and a Singlepower SDS (best sounding amp I've heard, though Singlepower & its owner went REAL bad those last few years) and Eddie-Current Zana Deux for tubes. There's new options I haven't tried (Beta 22, new Eddie-Currents, TTVJ Millet tubes/transformers) that should be spectacular.