I've had my HD600 for around a year now and used to pair them up with my fiio e7/e9 combo.
However I never really used my HD600s all that much since on the e7/e9 combo they sounded a little sibilant, cold and harsh to my ears, I wasn't too sure if this was due to the amp or the headphones(since the general consensus is that the HD600 is more neutral compared to the HD650 I thought it was the headphones). For extended listening sessions I would have to listen to tracks eq'd for extra bass in order for them to be comfortable. For added perspective I actually preferred listening to my pair of DT770 pros 80ohms over my HD600s, simply because the sound is more relaxed, though muddy and too bass heavy.
Anyway I replaced my e7/e9 in the last month with the Magni/Modi stack, and I'm happy to report that the sound has much improved! I didn't get a chance to A/B the e7/e9 directly against the Magni/Modi stack, but I instantly noticed an improvement on soundstage, clarity and most importantly bass punch+warmth. Overall the HD600s sound much more musical coming out of the Magni/Modi stack, I now find my ears isolating and appreciating the individual instruments in each song, e.g. individual guitar plucks have more warmth depth and texture to them, this wasn't really possible with the e7/e9. And now I'm happy with their sound with no eq straight out of itunes(and the DT770s get no time on my ears!).
In the interim period between selling my e7/e9 and buying the Magni/Modi stack, I started to use the HD600 unamped straight from my macbook and noticed that they were actually quite enjoyable and easy to listen to compared to the e7/e9. The sound was very muddy and had little soundstage but had a nice warmth similar to what the Magni/Modi stack has.
So to conclude I'm pretty sure that the HD600s with the Magni/Modi stack will blow your unamped HD555 out of the water. Additionally, although I'm not too sure how the Magni/Modi stack compares to other similar priced DAC/amp setups like the ODAC/02, if you can't find any conclusive evidence to tip you over any other DAC/amp combo in the price range, I can recommend that the Magni/Modi combos well with the HD600s to my ears!