What's best to look for here, both for a headphone amp and any amp in general with an output e-cap: lowest ESR, highest ripple current at low freq or high freq, lowest impedance, or just a cap that averages it all out?
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Simply, consider that the main power supply smoothing cap is also doing a similar duty in the power supply to what the output cap is doing for audio alone. You could do worse than use the same spec. cap there at the output, assuming the PS cap(s) is a decent grade for audio amplifiers and appropriate value in the first place.
You might want to take a look inside the high end signature and masterclass range A21a class A amp ranges by Sugden (UK) to see that's just what they do.
Sugden Audio A21SE - Neophonics(R)
You might want to take a look inside the high end signature and masterclass range A21a class A amp ranges by Sugden (UK) to see that's just what they do.
Sugden Audio A21SE - Neophonics(R)