I would investigate an external DAC to eliminate noise contamination from the computer. I'll post this and see what is available.
This is really beyond my expertise, but here are some points:
1) some are USB (so should act as the processing device as well)
2) some connect to the onboard audio digital output (TOSLINK) to convert that into an analog output (so your desktop speakers would use analog output, and the headphones would be using the digital output via the DAC)
3) Some DACs support desktop speakers AND headphones, some only one or the other (and may be limited to stereo)
4) Quality varies a lot. As does software support.
5) Again, DACs eliminate analog contamination to avoid "hissing" or other issues that are common to desktop headphones. Even a good sound card has a hard time eliminating this.
6) *If you get a good sound card, not a DAC, make sure it can connect to both your speakers AND your headphones. Or you can connect the headphones to the speakers if that's supported but the sound quality would be affected (hard to say how much though).
7) A good sound card might not sound noticeably better than a recent audio chip in a modern motherboard.