Thanks all for the responses. It's sharpened my thoughts on this a lot.
Promotion is notoriously hard in my department. They won't do anything off the back of shortlisting. You have to get an offer, then they will move (even then our HoS has been known to ask women who've been offered Chairs elsewhere whether this is serious because of the expected commute!). Unfortunately I can't speak to the other institution's old HoD because he died unexpectedly shortly after leaving.
I know that a job offer won't be magicked out of thin air, and of course I expect there to be an open competition for any position but what I was hoping to achieve is to get myself on the radar of the new head of school and be seen as the natural next appointee. I am ready to go if that's what I'm told. I have been exploited plenty in my current position so I'm not feeling much loyalty at the moment. I have no problem with this getting back to my current department. It's very clear in my institution that this is how it works - trying to be a "good girl", working hard, and being promoted because you are good at what you do, doesn't cut the mustard here, sadly. I've tried that.
I don't think it matters that I was unsuccessful last time (although fwiw, I was told that I was the department's first choice after the research presentations). Since then, there's been a new head of school, with a very different focus, the two senior people in my field are even closer to retirement, and the person who was in the equivalent position to me left under a cloud. The two junior academics are more in a related field than my actual one and have been drafted in to cover teaching shortages. They don't research in my field. So they are short in my field now, and will be shorter still in the near future.
Things I think make me attractive are:
Good publication record eg I have seven pieces out/coming out this year which I think is pretty good for humanities.
I established a research centre a few years ago which is unique in my field in the world. I have essentially brought to my discipline a whole new lens of analysis. I am and have been approached by other institutions for collaborations and we have been successful in bringing in fairly large grants (not science level, but in the hundreds of thousands). We have really good connections and (financial) support from industry (and we are the only centre in my department which has achieved this).
Colleagues at this other institution have encouraged me to approach their new Head.