Francois, thanks for the links, but I want to get a fine tuner that fits on a normal tailpiece just for the E string (in my case, a long C).
And no, Nick, there's no problem with the extension at all. It's just the nature of extensions that most of the time, most of us play them with one or more of the capos closed. Thus, in order to make minute tuning adjustments in the course of a performance or rehearsal, we are obliged to first, open the capo or capos (four in my case), retune, then close the capo(s) again.
Whereas, with a fine tuner, you only need turn the knob in or out a tad, and Bob's your uncle. No need to open and close capos to make the adjustment.
I've had one on my principal bass for decades, and can't imagine playing an extension bass without one. Now I want one for my backup bass.
The problem is that no one seems to sell them any more... Or do they?