I don't think your thoughts (and probably that of most headfiers) on having a secondary or complementary headphone are restricted to the T1. As good as any headphone may be, for reasons of different genres of music or simply our mood at the time, it is good to have an alternative sound signature to perhaps our preferred/primary home listening headphone. For me, I found the LCD 2 to be a great complementary headphone to the T1, as they both have great sound signatures albeit quite different from each other.
Dependent on your requirements, it might be an opportunity for you to consider a closed headphone as an alternative, thus providing you with an option for isolation from the environment. I have the the Ultrasone Signature Pro which I find is of similar sound quality to the T1 and LCD 2 but with a signature that is different to both. If you didn't want to spend close to $1,000 then there are several good/great sounding closed models for around $300-$400 including the B&W P7, NAD Viso HP50 and PSB M4U 1 and 2 to name a few.
If you want to stay with a traditional open back home headphone to complement your T1 it is a matter of selecting something that has a different signature. Maybe a Senn HD 650 or a Grado RS1. Actually scap the Grado, as you find the T1 a bit intense. Not a word I would have used in describing the T1.
I guess the question to ask yourself is:
When you are not in the mood for your T1 what would you like to be different?
