There are a number of headphone amps in the ~ $300 range that would offer benefits vs. the headphone out in a receiver or integrated amp.

I use a Musical Fidelity X-CANV3 with HD580s. Very smooth, very liquid sound with good definition. Not the last word in resolution, bass extension or treble extension but that might be because I have used no headphones other than the HD-580 I've had for many years.

None of my components offer balanced connections so I have no comment there nor do I have any experience with Grace and Headroom components save they have received many favorable reviews.

Like Tevo, I use the XCanv3 with 600's. A surprizingly powerful upgrade I made is the Cardas replacement cable. Plug it in and things get materially better. The cable added about $200 in cost. Other headphone amps - with much higher prices - might produce still more improvement, but I'm into the value thing, and would rather spend the additional money on music.

I use 650s with an Equinox cable (I find the Cardas may be too bright for my tastes; I was curious about the Zu cable, but, since the consensus is that it's brighter than the Cardas, I've passed.) I have a Grace m902 and usually use the crossfeed circuit. It's fairly subtle and shifts the tonality a little, but, creates an ease that allows me to listen for many hours at a time. I don't care for the m902 (or 901 - I own both) onboard DACs (a bit grainy sounding.) The Grace amps are best used as amps only. Considering the number of features the m902 offers, it probably is a bargain. But, it doesn't have a great DAC and you might do better if looking for a smoother sounding amp. If spending the same money, I suspect you might get better performance from the newest series of Headroom amp(s) with crossfeed. I've only heard these amps a couple of times, for brief periods, but, they're smooth and have a relaxed, natural quality that is appealing. Their crossfeed circuit is probably more sophisticated than the simple Jan Meier one that Grace licensed.

Generally, you'll get much better performance from a dedicated headphone amp. The headphone circuits in integrated amps often use a cheap op amp to drive the cans. If you're worried about budget and are handy with a soldering iron, you might even be able to squeeze some better performance out of the NAD before you upgrade. If the op amp in the headphone section is an older TL072, TL082, or NE5532 dual op amp, you could pop a smoother sounding, more responsive Burr Brown 2132 or 2134 op amp in there.