I have the AKG-701 and have heard the Alessandro MS1 (decent listen) and the Grado SR-325. I like hard rock/metal, as well as Goa Psy eletronica and other dance stuff like Oakenfeld and Van Dyk. I also like my movie scores too tho
Anyway: IMO, the bass on the headphones is particularly important for you, since you're listening to bass demanding music now (not bass *heavy*, just demanding). With the GLite/DPS, the K701 has deeper bass extension than the HD560 (quite noticeably deeper, with the 650 just being warmer in the midbass), and both these cans had deeper bass (when amped, I couldn't say for sure unamped) than either of the Grado/Alessandro models I've heard (although, they're not the top of the line models).
I've had others argue these observations, I just point them at the Headroom graph which altho can't be taken as gospel, lines up exactly with my opinions. Must be coincidence
And I hear the deeper bass of the AKG's on a virgin K701s. When they are burnt in, the bass gets a little warmer (but still as tight) and the mids sound closer to the HD650 (albeit not as prominent, since the rest is more balanced IMO). I changed from the HD650 because even with an amp, it wasn't *quite* nimble enough for my metal/eletronica. If I had to guess (and I am), I'd say this could be the change ur looking for too.
I would leave ur current source and get an external DAC. This made the next biggest difference to me after finding cans with the sound sig I liked. If you like your DT880, stick with it! Otherwise, I would get the K701 and after it's burnt in (200 hours will safely do it), then compare and see whether u like it more. Then sell the one u don't need (if u need to... could keep both). Just my2c.
Or alternatively try some of the cheaper Grado's to get the "taste."