For those interested:
TIED (TIDE, DIET, EDIT)
According to the Internet Anagram server, there aren't any more beyond that.
My web search didn't really turn up a name for these. Someone called them word chains, but that was the only place where I found them called that, and it was on someone's Scrabble page. Now, do I write a program to find these in both directions? Or do I spend my time more productively?