I think some PBS stations do air Charlie Rose in the middle of the day, but for most of them, all other programming from 6 am until at least 6 pm is children's programming. I actually have 4 PBS stations on my cable system here. Of course I have WGBH and WGBX from Boston, but I also have WENH from New Hampshire, and WSBE from Providence. WGBH dropped the afternoon airing of SS in October of 1996 with the premiere of "Arthur." And I think the same goes for many PBS stations throughout the country. WENH had SS on twice a day until February of 1999, and it was replaced by Zoboomafoo and Zoom. WGBX did away with their single airing of SS then, too (though they brought it back later that year, and then took it off again in 2001). WSBE had SS on twice a day until 1994, when they only had it on in the morning. And in 1999, they actually moved it from the morning to the afternoon. Then in 2002, they moved it back to the morning. But see, the thing with WSBE is, up until 2002, their selection of children's programming was pretty limited. Here's how their children's schedule was from September 1999-January 2001:
3:00 Teletubbies
3:30 Sesame Street
4:30 Zoom (replaced by Clifford the Big Red Dog in September 2000)
5:00 Zoboomafoo
5:30 Arthur
6:00 Barney & Friends
But believe it or not, they still air in-school educational programming during mid-day. And I think some of those programs are quite old, believe it or not. I flipped by that channel once like last year or so, and I actually saw the PBS ID from 1989.
There are a few PBS stations that still air SS twice a day, but not nearly as many as before. The one that I find most unusual is WITF-33 in Harrisburg, PA. They currently have SS on at like 9am or 10am (one of those), and then it repeats at 7pm! I remember going there back in 1993, and I remember they had it on at that time. But I spoke with someone at the station recently, and they said that it has always been a part of the community, and it gives the opportunity for parents to be able to watch it with their kids when they get home from work, and it does very well during the 7:00 hour, when not many kids shows are on. I remember hearing the same thing about WSBE airing Barney at 6:00. I think they had it on at that time from the time they first started carrying the show in 1993 or so until about maybe 2001.