Okay, I rewatched the Daily Show segment with Jon Stewart hammering Bill Bennett and I see that I misunderstood: Stewart did not say Bennett would be on the following broadcast, as well. What he did was tell Bennett, "We're going to split this into segments…and you'll be the whole back end of the show." He did not mean the interview was being divided over two programs. He meant it would be edited so it could run two segments in that night's show.
What I think happened was this: The show has three acts. I believe they taped comedy spots for the first two acts and then the Bennett interview was supposed to just be Act Three. During it, Stewart and/or his producers apparently made the decision that it deserved more time so they let it run long enough to fill two acts. Then they went back afterwards and taped Stewart welcoming us back from commercial. Afterwards, they dropped out whatever they'd taped for Act Two and positioned the Bennett interview as Acts Two and Three, using that "welcome back" to get us back into the interview at the top of Act Three. That caused the awkward edit right after the "welcome back," and I think there was another jump cut in there, probably also because they were reconfiguring one segment to span two.
It was an interesting interview, in part because Bennett's a smart man and he knew he was losing badly. You can see portions of it over here.