If you can, check out an old video/dvd with Pat Page on the TT, some great ideas and sound advice on handling.
I think L&L have it? Possibly a Trik-a-Tape production?
Years ago, when the TT was exposed on TV, I used to do an effect where I vanished a Red silk with the aid of a pull.
All the time I wore the TT and at the end someone would invariably comment on my 'unusual' thumb. So I would pull it off and show that I had a white silk in there, "No, that's my hanky".