You can build a CMoy with an AD8397 as a direct substitute for whichever opamp. You will need to etch a board, however, to take advantage of the high current/power of which it is capable. It needs the pad on the bottom soldered to a large area of copper on the PCB to act as a heatsink. The easiest way to do that is to use SMT components on the top side, leave the bottom surface unetched and drill a fair-sized hole that will allow you to fill it with solder and make a connection between the bottom pad and the copper. You can surface-mount regular components too. Using regular leaded resistors means you can use them to jump over tracks so that with a bit of messing about you don't need to drill any holes. Read what the datasheet says about layout considerations.
The CMoy circuit will work, with the provisos I made about layout and heatsinking, but adding short circuit protection will degrade the amplifier performance or add a lot to the complexity of the circuit.
If you can trust yourself not to plug or unplug the headphones while the amplifier is turned on, or use connectors which do not short the output when connecting or disconnecting you can use this chip. You could use RCA connectors with a RCA to TRS jack adapter. As long as you only disconnect the RCAs from the amplifier as opposed to pulling out the TRS, you should be able to avoid shorting the output.
Okay, now how about a different scenario? Suppose the AD8397 goes on a ready PCB with ceramic capacitors (all according to the spec/recommendations of the datasheet) and is used on a DAC line output. It will have to work with a TRS headphone socket, and hence will need some kind of short protection.
The issue with AD8397 is, it's one of the few headphone amps which match the SNR of AK4556 (~100 dB), the other one being AKM's own AK4201, which unfortunately does not work with high-impedance (600-ohm) headphones (think K-240 Monitor, classic K-141, etc.). The AD8397 has the right spec (+/-12.6V is within 600-ohm voltage) hence it's about the only thing which is suitable.
Why do you say that the AK4201 is not suitable for driving 600ohm loads? I suppose the datasheet is somewhat incomplete compared to others in that it does not have a THD VS load impedance at a fixed output level, even separating 150ohms & up from 150ohms & below.... BUT IME there are very few op amps that can drive 16ohms that cant drive 600ohms. Maybe not to 120db and maybe not with 3EE-15% THD, but um, its a portable - accept that some compromises will need to be made.
What's needed is two designs really: one a simple battery-driven amp; the other, a headphone amp stage for a USB DAC based on AK4556, with step-up USB power. Both using AD8397. The pocket amp is more of a curio for the sake of it, the amp stage is a bit more serious and will be based on a PCB. Wakibaki has a schematic of an interesting AD8397 design at his site, so that might do it.
Have you applied* whatever you are measuring to an actual headphone to determine what you need, or is that 6-12v someone else's number? I have heard as high as 45 (what units, I dont know, I'm quoting someone else's number) is necessary for 600ohm headphones.
*Start off at low levels and work your way up slowly. People have destroyed headphones with less than 5V of current before. Pay attention to clipping.
The AD8397 comes in two flavors, the -ARDZ and the -ARZ. The -ARDZ has the E-pad, which allows you to use the PCB as a heatsink for better cooling for higher power applications. In the Mini3 the -ARDZ can be replaced with the -ARZ, see Mini3 construction notes. My Mini3 has the AD8397ARZ in it. Works fine too.
The pad doesn't need to be soldered in order to function, the Mini3 construction notes advise to apply some thermal grease between the pad and PCB. Some applications can use the ARZ version.
Mini3 prototype PCBs were made so you could solder the E-pad to the PCB using a through-hole under the E-pad. Mini3 production PCBs do not have the through-hole. I guess a soldered E-pad makes removal of a faulty 8397 one hell of a job.