Thanks for the responses, folks. It seems like a headphone amp will benefit me greatly, but I'm still wondering if I should buy better headphones first. I did a little experiment this morning and plugged my Senns into my Macbook (I'm not a computer audio guy yet) with its output impedance around 30 Ohms, and despite the crappy sound card the sound was worlds better than it was through my NAD. Still not a great impedance match, but it was better.
Though the bass was diminished when plugged into the laptop (I think the impedance mismatch with the NAD emphasizes the mid-bass because of the Senns' high impedance spike around 100Hz), the mids and treble were clearer. Cymbals sounded like metal rather than papery electronic hash, though resolution still wasn't excellent. A brushed snare on Lucinda Williams' "Fruits of My Labor" sounded like vague background clatter (think paperclips being shaken in an empty OJ-concentrate can), while it resolves perfectly through my speakers. Not sure if that's a limitation of the headphones, though I do see that they're capable of better sound than I was getting.
The Portal Panache is new to me and looks like a great product. That might be a good future upgrade for me. For now, it's a little out of my reach. I've read a bit on Head-Fi but haven't joined yet, so I appreciate everyone's input here. Still don't know if I should put the Schiit or headphones on the list first (ideally I'll eventually do both)--but I'm open to further suggestions! Admittedly I'm still not a huge headphone fan, as listening fatigue sets in pretty fast for me even at low volumes, but they're good for keeping peace with the neighbors.