Well, I'm actually going to answer my own question.
I can do that, because I recently saw both Warren Haynes and Joe Bonamassa play. I was close enough that I could actually see for myself how they do it.
So even though I like this question to be quite general (in order to be helpful to others), to be honest, I actually just wanted to know how Warren Haynes does this ;)
And I can say that Warren actually puts the pick between his index and middle fingers, high up, so he can still move all of his fingers to some degree.
In that way, he is able to finger-pick with all of his fingers, and not sacrificing any particular finger. The trick seems to be though that you can't go crazy with the finger picking, or the pick is going to come loose.
As well, you have to really practice that transition of moving the pick from your normal grip, to holding it between your fingers, and vice versa. Warren does it smoothly, and without effort. He's obviously been doing this for decades. I think it's going to take me some time to get even close, but I'll keep trying.
The other benefit to Warren's technique, is that as a very good slide guitarist, Warren naturally has very well developed finger-picking technique with the thumb, index and middle fingers. (Warren's style of slide involves plucking and muting the strings with those fingers). So of course, it then makes sense to keep those fingers free when finger-picking with the plectrum tucked away, to make the most of that well-developed picking technique. I was thinking that when I originally wrote my question, and I can see that it is true.
The technique Bill Cheatham describes for Brian Setzer is actually the same technique Joe Bonamassa uses.
I'm going to concentrate on Warren's technique however, because it is closer to what I'm hoping to achieve with my playing.