If a person has been absent under the 52 week rule and then remains absent but doesn't come under the 52 week rule any more, can an absence that comes under the 13 week rule start directly after? Say, for example, a claimant leaves hospital after a 15 week stay and goes to stay with a friend for 10 weeks.

The short answer has to be no. When coming out of hospital they would have to return to the home first before going to stay with the friend. Otherwise the "intention to return" was demonstrably lost when they failed to return to home after hospital.