A lot of audiophile terminology is nonsense, and there is a ton of snake oil in that industry. Like, a truly comical amount.
BUT, not everything “high end” audio people talk about is fiction. It is true that many specialized headphones require more voltage (or in some cases, usually high end IEMs, lower impedance) in order to drive a suitable volume without distortion.
Smartphones often don’t have enough voltage output for headphones which were designed around the output of studio equipment. If you don’t have enough voltage then your headphones will clip when you increase the volume past a certain level.
Clipping is a measureable phenomenon, not a subjective (made up) audio property like “soundstage.”
Anyway, how much voltage you need is something you can calculate for your headphones. An HD650 needs ~1.8V to be able to handle full dynamic range without clipping. Your AT headphones probably need more like 1V. The adapter does 0.44V, whereas the headphone jack on previous iPhones was about 0.5V.
That 12% decrease isn’t a lot, and it wouldn’t impact any headphones sold as portable consumer units, but in your case it might be enough to increase clipping to the point where it becomes annoying at the volumes you generally use. My guess would be that you were right at the line before, and now you’re on the other side
Unfortunately what you need is really a pair of headphones with a lower impedance (or higher sensitivity, or both). Wish I had some more helpful ideas, but it’s a physics problem and that’s just how the numbers work out.
I should probably point out that I listen to a lot of ambient, orchestral, and live music. For the average rock song I'd be hard pressed to tell the difference.
That’s definitely a major contributing factor. The music you listen to has a wide dynamic range, so you need to crank the volume to hear the quiet parts (modern studio music does not have quiet parts) and anything loud will clip because of the voltage constraints.
Some reading for anyone who cares:
On power requirements for headphones. Written by the engineer who was famously banned for life from Head-Fi by bringing actual science into the conversation, thus instantly discrediting most of the products in the audiophile world:
http://nwavguy.blogspot.com/2011/09/more-power.html?m=1
innerfidelity.com takes measurements of the lighting to headphone adapter, compared to other Apple devices with headphone jacks:
https://www.innerfidelity.com/content/apples-lightning-headphone-adapter-analog-or-digital
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TL;DR:
Voltage limited due to the high impedance of your headphones, you will get distortion at high volumes, nothing you can do other than replace the headphones or use lower volume setting.