I have heard every single headphone you guys have mentioned except for the HE-4. If you want a better DT880, the T1 is without a doubt the new DT880. The HE-6 is also very similar to the T1 except with better mids. Slightly different sound but better to my ears especially if you have the amp to drive it. The HE-500 isn't as similar to the DT880 as the treble is a bit too "tamed" in comparison. HD600 would be wayyy different from a DT880 sound - smooth, really smooth, too smooth with super tamed treble. HD800 would be the most extreme deviation sounding quite thin and just plain different.
Go with a T1 if you like the sound of the DT880; it will be a significant upgrade with almost the same sound signature. Go with an HE-6 if you want a significant of the DT880 but want neutral to slightly forward mids version.