Even though the HD558 is a tad better than the HD555, they are IMO both great headphones. (though my experience comes mainly from reading reviews and watching graphs)
Onboard audio however can suffer greatly from interference. Besides the built in amplification quality there's a good chance the audio circuit on the motherboard receives interference from other power circuits that are nearby. Expecially as there are massive amounts of power running through the motherboard between the videocard and processor.
I heard good things about the Asus Xonair. It is supposed to have a pretty decent headphone amp built in, and it won't suffer (or suffer less) from motherboard interference.
You can however also look at an external DAC/amp. For under €100/$100 you can get a Fiio e10 USB DAC/AMP. I am not sure, but my guess is this would give you the best sound quality*, but would limit you to stereo output and an external box.
*I don't know the impedance curve of the HD555 so I am uncertain if the e10 has enough output power to deliver bass for the HD555, but it is worth investigating.
Good luck!