I use one of the tiniest models available: KINGMAX Super Stick . Other manufacturers also have similar models. And there are card-readers for micro-SD of similar size.
The size and weight are extremely small — perfect for EDC, but the problem of finding the right place for it had to be solved. So I looked at stuff I always carry with me and found a fold in my wallet.
The last ingredient — velcro to attach the drive. For such a tiny drive it more than enough.