Integrated amp w/decent headphone out under $5k

For headphone listening I'd go with an amp designed as a headphone amp. Frankly, for many integrated speaker amps the headphone circuit is an afterthought. 

I'm quite fond of the Schiit headphone amps. While my Sundara phones aren't as fancy as yours, I'm listening to them while I type this response, using the just recently released Schiit Midgard headphone amp with their new "Halo" circuit that incudes the headphone drivers in the amp's feedback circuit. Have to say I'm impressed. And the price is a near giveaway. Schiit has a number of fancier models.

And Hifiman has a number of their own dedicated headphone amps for you to consider.

Why do you want an Integrated Amp? Do you have speakers? What are they?

I like everything about my SST Ambrosia.  Great phono and headphone amp

@leemaze yes - other equipment listed in the initial post and also my virtual system (click on my ID)

@timlub Never heard of SST but interesting product. Out of my price range and looks like a preamp only - appreciate the info