What should I upgrade?

The HRTs are very very good, a real value leader, but given your budget constraints, you might want to consider a DAC/headphone amp combo. If you like the Senn sound, the 580s are considered very nearly as good as the 600 or 650 and much cheaper.

I found AKG K701 step over the performance level of Senn650.
They're the only ones I could listen for long long time.

But those AKG 701s need some power.

I would look into upgrading your headphones and getting a headphone amp. There is a $99 Schiit amp that might work well with headphones that aren't too demanding (maybe check at Head-fi to see how the 701s pair with the $99 Schiit).

AKG 701s are very good, for sure. My thinking is that OP is on a budget, likes the Senn sound. The 580s (and 590s I think) are considered very very close to the 600s and are much less expensive. That might enable the OP to get a headphone amp, as well, which I think would also be very important for upgrading his system.

@Swampwalker
I am not certain that I am partial to the Sennheiser sound, it is the only pair of decent quality headphones I have owned. I do think they sound good, but don't have much to compare them to. They are a hand-me-down from my father who used them for his studio recording stuff, they had some major damage to the plastic that holds the adjustment band, but performance wasn't affected, it was merely cosmetic.
I have been looking for a better solution, but I am not sure about the open headphone design, the passive noise canceling the HD280s have is probably essential to my listening experience, but that said, I don't have much experience with other solutions.

@Davide256
I would hate to ditch the Xonar Essence STX card if I can avoid it, I already did spend quite a bit of coin on it, and thus far I am quite impressed with it. I have used Creative (Crap-vative) cards and M-Audio, among other cards previously. It is surely in a class far beyond. As for the jitter -- I have heard it can be an issue if you are up-sampling a signal rather than running it at a native rate but I tend to run native. The solution you suggested also is not capable of 24/192.