I would say that the Tubescreamer would be a good play in this case. Fender Twin Reverbs are great amps for clean tones. They are just very hard to push into overdrive without getting VERY loud. The lack of a master volume doesn't help you out in that case...

There are tons and tons of overdrives out on the market. The Tubescreamer just happens to be the most widely used one. I'm sure Johnny Winters has one or something very similar on his pedal board.

You might want to check out the MXR Custom Bad Ass Modified OD. I have one on my pedal board and it's a versatile little pedal. It does the same thing as a Tubescreamer but it has some nice EQ options that allow you to shape your tone a little more. I am pretty pleased with it. It's also pretty inexpensive. I don't use a lot of OD... I'm more of a Fuzz guy but I like that pedal quite a bit.