I sort of made this point obliquely on the other blog, but one of the reasons I like the American version of The Office so much is that I find it gives me a lot of what I used to find in King of the Hill. KotH in its prime years had the same showrunner, Greg Daniels, and one of its best writers was Daniels' brother-in-law Paul Lieberstein, who now writes for The Office as well as playing the part of Toby. What they've brought to The Office is very much what they brought to KotH: realistic, observational humour with a satirical edge but with genuine emotion, some dark undercurrents (on KotH the darkness mostly came from the character of Bill, the most suicidal depressed loser ever seen on a mainstream sitcom) and more story arcs and status-quo shakeups than you'd get on a normal sitcom. KotH is still on, but while it's still a good show, it doesn't have much of that any more; the episodes are less satirical, there's less emotion, and the characters have become more static and unchanging. So for what I looked for in KotH, I now turn to The Office.
Anyway, the point of this is that I think it's actually quite common to turn to a new show to "replace" an older show, particularly if that show's been canceled recently. It doesn't necessarily mean that the new show has to be the same as the older one, or even very similar; after all, The Office isn't really similar to KotH in any obvious way. It just means that sometimes a show will come along that has certain qualities that a) most shows don't have, and b) used to be provided by a show that doesn't provide it any more. When that happens, the fans of one show will sort of migrate to the newer show.
For example, in 1978 it was pretty clear that Taxi was successfully making a play for the Mary Tyler Moore Show fans; the shows weren't similar in setting or characters, but they used a lot of the same creative talent and had a similar mix of highbrow and lowbrow comedy styles and a similar attitude (though Taxi was, at least at first, a little darker). Then in 1982, when Taxi was on its last legs, its fans migrated to Cheers, which not only had a lot of Taxi people working on it but sported a similar cast of lovable losers on a huge single set.
Or take Family Guy. I'm no fan of that show (as you may have heard rumoured) but it definitely has pulled in a lot of people who became disenchanted with The Simpsons and were looking for another show to give them the fast-paced comedy and irreverent attitude they loved on The Simpsons in the David Mirkin years.
Are there any shows that you turned to for the particular pleasures/qualities you used to get from another show?