The tape out is a line level output that bypasses the amplifier and volume controls of your integrated amp. This should allow for input into a headphone amp. The headphone amp will control the volume for your headphones.
Headphone amps are like all other high end audio equipment. They each have unique sonic character and in general sound better the more you spend. Headphones also have very large differences in power requirements… just like main speakers. I have owner headphone amps from a hundred bucks through my current setup in which with high end 300B tubes cost over $6K… and worth every penny in sonic character. This has enormous current capacity, so any headphone is going to sound it’s best. It is a Woo WA5… you can see the system under my user ID, virtual system.
Many good headphones are hard to drive. So, it is important to match the headphone amp with
I highly recommend looking at Woo headphone amps. They are incredibly musical and natural.