Again this is just my opinion with my years of experience - but the secret to getting headphones to sound good has nothing to do with Axe III - other than making sure you are outputting in Stereo vs Summed MONO - and making sure your internal output isn't clipping. It's the headphones. Specifically - it's mainly the OHM rating on the headphones - what you need are something that can handle the volume of the output (not too low) but something you can still push (not too high) without a headphone amp. I'd shoot for something that is about 32-50ohms. I have a pair of 300ohm headphones and I never use them because it takes a lot of power to push them and I don't need extra complication in my life. If they are rated too low, when you turn up the volume they will crack and distort. That's the main thing.
The rest is your pref in headphones. Some people like closed vs open back etc, etc... I won't get into all that. My top recommendation would by anything 50ohm or near 50ohm Sennheiser - their stuff just pleases me to no end in terms of sound and comfort. Alternatively you could go with Beyerdynamic DT 770 Pro 32 or 80's.
For in ears I use Shure SE215. They translate live perfectelly from my Sennheisers.
The short answer is - you don't have to do anything to get your stuff to sound good - just get the right headphones and it will sound amazing. In fact, if your listening experience is similar to mine - I'd argue you aren't really hearing what this thing can do unless you are using headphones.
I will mention one setting - in the cab block there is a air or space parameter like the 3rd or so tab down that will make you have that improved feel with some amps when using headphones. Sometimes I use it, sometimes I don't. Just preference.