Lots of great suggestions here.
I've been wanting to try a tubed headphone amp (most likely Woo), but haven't experienced one, and hesitate to make recommendations for products I have no experience with.
Your HD650s are really nice and will benefit from a better cable and a good headphone amp. I have a pair of HD6xx, which are basically the same and replaced the stock cable with a balanced cable and use them with a Sony TA-ZH1ES amp. I have a dozen headphones including some TOTL cans like Audeze LCD-4, but find myself going back to the HD6xx for their comfort and how easy they are to listen to.
I have also replaced the stock earpads on most of my headphones. This doesn't typically result in a big sonic difference, but almost always increases comfort.
For higher end headphones, the Sony MDR-Z1R are a great value. Excellent sound quality, deep bass, while being light and comfortable.
Note that I tend to focus on comfort as much or more than sound quality. The LCD-4 sound amazing, but they are heavy and not the most comfortable thing to have on your head for a long period of time.