in my experience classical has to be played through speakers and headphones which have great imaging, plenty of detail, and a sweet midrange, otherwise it is very difficult to stay into it (for a hilarious description of the "spacing-out" effect watch the classic talk of Benjamin Zander on TED http://www.ted.com/talks/benjamin_zander_on_music_and_passion.html).
Of the many good headphones mentioned in this thread none really fits the bill, unless you connect them to a well-matched head-amp. So you are looking at a good $500, perhaps $400 if you are patient and lucky in the used market. My top mid-fi recommendations would be a used AKG K501 which would work OK off your Bithead (I owned the combo). It will sound unbearable with less than well-recorded music, but luckily classical music is almost always well recorded. The K701 and Senns 600 are superior technically but (much) harder and hence expensive to drive. If you are ready to spend that money than I would rather buy a Hifiman HE400 and a decent amp.
If you need a quick fix that would work well un-amped, I would consider the (gasp!) Skullcandy RocNation Aviators. Only yesterday I spent half an hour listening to the reference-quality MTT/SFSO recording of Mahler 1st Symphony on a iPod Nano driving the Skullcandy and I could not believe how good these little closed thingies were at imaging and detail extraction. They are way too warm to be your ultimate classical music headphone, but they are definitely good enough to keep you engaged in the music.