You're better off with a dedicated, separate headphone amp. Most integrated amps and receivers run the speaker outputs through a 220 to 330 ohm resistor to the headphone jack, and thus do not provide an optimized output. The high output resistance will cause coloration and distortion (I have seen this personally with my own tests). Very, very few of them contain a discrete headphone amplifier. The additional benefit is that you can upgrade at will.
Integrated amp w/decent headphone out under $5k
Could use some help picking an integrated amp. Current Eastern Electric M520 24W EL34 doing a nice job driving my 95db efficient Omega CAMs. Would like to get to next level of tone and refinement and also drive my Hifiman HE1000 Stealth headphones (supposed to be easy to drive). Tried a SET separate but missed the bass definition and dynamics so I think I want to stay push/pull or warmer sounding solid state or hybrid. 10-25 watts should be plenty. Current leading candidate is LTA Z10… anything else I should look at?