The problem in your design isn't the output transistors and neither will this be solved by using darlingtons - in fact the problem will become worse with darlingtons. Look at the spec of the NE5534 and read the section about maximum output voltage swing.
On page 5 it tells you that the max swing might be as low as 24V on a +/-15 volts supply i.e. a 30V supply only guarantees an unclipped output swing of 24 volts p-p - that means the internal transistors in the device "lose" 3 V top and bottom. This doesn't magically improve on a lower supply and you should expect, on a 9V supply, to generate a maximum output swing of 3 volts peak to peak.
OK, if you used the typical spec of the NE5534 this might rise to 5 volts p-p.
This is then compounded by the push pull stage - it's an emitter follower push pull stage and it's output will "lose" between 0.6volts and 1 volts per transistor depending on your load.
This leaves you with a crappy 1volt to 3.8 volts p-p being fed to your headphones before clipping takes place. If you use darlingtons, they "lose" double what a regular transistor loses so this will be even worse.
Recommendation - use a rail-to-rail output op-amp. There are plenty around these days and they'll probably only drop about 100mV per rail.
As mkeith says, it may not sound as clean as a linear amplifier, and you have to remember that headphones are a lot more sensitive and can be a lot more critical of low level distortions than loudspeakers.
However if you still want to use it, here's how.
Realise that 3W is waaay more power than you need for headphones - so is 0.375W. I'd go further and say that these power levels can be dangerous. 115dBa through speakers is wall-shakingly, cup-sliding-off-table, stomach-hittingly loud. Loud.
The same level - measured with B&K SPL measuring apparatus) from headphones - without all these other cues - sounds ... well, a little bit loud. And that's probably under 0.1W with many headphones.
In either case, the Health&Safety Executive (UK) recommendation of 90dBa safe exposure over an 8 hour working day, to protect your hearing, translates to listening to 115dBa for ...
nearly 4 minutes.
Without all the other cues, headphones can leave you quite unaware of the danger. The fact they go so loud and stay clean and undistorted doesn't help.
(Still, the rock radio studio techs wanted them LOUDER!)
So it's OK to use this amp below its full power rating.
Now - the PAM8403 datasheet doesn't say what's actually inside it, but it's a reasonably safe bet that it's a full bridge for each channel. As Out_L+ is driven positive, OUT_L- is driven negative, and so on, giving you twice the voltage and 4x the power that a half-bridge will do. (With no input signal, both outputs will sit at half the supply rail ... check this with a meter).
So I would simply ignore OUT_L- and use OUT_L+ as a half-bridge (a normal push-pull amplifier). You will need to AC couple the output via 10uF (maybe 47uF would extend the bass a little) but then you can simply return the common GND from your headphones to GND.
simulate this circuit – Schematic created using CircuitLab
It can't damage the amplifier and if you have all the bits it costs nothing to try.
Best Answer
A transformer to make 600 ohm headphones appear as an 8-ohm load for the amplifier may be a bad idea...
Low-impedance headphones are ubiquitous. Plugging one into your transformer would present a load of very low impedance to the amplifier, which might dangerously overheat.
Most audio power amplifiers are designed for an 8-ohm load - some for 4 ohm, and a few for 2 ohm. Any of these won't complain if they see a load of 600 ohms. It is difficult to say if you'll get enough volume from your 'phones before distortion sets in. Those 600 ohm 'phones might just give you adequate volume when driven directly - no transformer.
It comes down to Ampere-turns. Drive an 8-ohm speaker with the same voltage as 600 ohm headphones. The 8-ohm speaker gets more current through few turns. Headphones get much less current through many more turns. I have 4000 ohm headphones that sound about as loud as 8-ohm headphones when plugged into an amplifier designed for 8 ohms.
So just try any of the amplifiers that are designed for 8 ohm load.