Before you shell out $200, $400, or even $800 for a dedicated headphone amplifier, decide what it is that you want from such a purchase.
Pride of ownership, exclusivity, bragging rights, being a member of the "in crowd", are all valid reasons for buying any consumer product be it audio equipment, automobile, or designer clothing.
If, however, your purchase decision is based on a desire to further explore the full audio potential of your recent purchase of the HD-600 headphones, consider purchasing a previously owned stereo integrated amplifier or stereo receiver.
I'm not suggesting that you need to start out by purchasing a vintage all-tube rig to experience the best possible sound from your 600s. Solid state, discrete transistor (no integrated circuits or op amps need apply) audio amplifiers and receivers from the late 1960s, all of the 1970s, and up to the early 1980s offer an incredible bargain waiting for today's savy headphone enthusiast.
Instead of struggling to match the power demands of the latest dynamic headphone designs with the puny power output available through the headphone jack of most of today's crop of audio gear, consider a modestly powered stereo receiver, with say 20 watts or so of true 8 ohm, RMS FTC rated audio power from the 1970s.
You want bragging rights, ok, buy a vintage piece of Marantz audio equipment from the early to mid 1970s. The higher powered Marantz receiver models such as the model 2270 are well known for their conservative power ratings and superior sound and attract a significant following, and not surprizingly command relatively high used equipment prices. No matter, the lower powered 2215, 2220, and 2230 Marantz receivers from the same period are largely unappreciated and can often be purchased online for a bargain price. If you want to see for yourself, check out the prices on eBay.
Other high quality, but lesser known and/or underappreciated brands of transistor-based audio equipment from the same period include Fisher, Harman Kardan, Sansui, Scott, Technics, and Yamaha.
Many of these fine old receivers fetch less than $50, sometimes way less. Even factoring shipping and insurance, your final cost should be far less than $75. Often these units have scratchy volume controls. This can usually be fixed with judicious application of one of the better cleaner/deoxidizer sprays. Sometimes one or more of the front display lights are not functioning, usually replacement bulbs are available. If you are going to spend any time listening to the FM tuner section of one of these vintage receivers, and you probably will, you should take the unit to a good repair shop to have the FM tuner realigned. This simple repair typically costs under $50. Think of the FM stereo as a bonus.
Do some research, read up on the vintage equipment on the net, then hit the auction sites and used equipment sites with the knowledge that you are going to score an unbelievable bargain in headphone amplification. I'm not talking about a headphone amp for portable use, for that application there are some very good current models and plans available.
If, after auditioning your HD-600s through the vintage receiver of your choosing, you don't like the sound, all you have spent is something less than $100. But I predict that once you have heard your HD-600s when connected to a vintage Marantz or similar unit, you will have found what you are looking for in superior sound without having to spend much cash.