More control of the summary section

I posted Issue 1495 the other day and was pointed here. So, feature request:

It’s great that I can use <!--more--> to set the post summary (with markdown), but if there’s some content at the start of the page that I don’t want to include (like a ## section title) there’s no way to skip it.

It’d be nice if I could denote a section of text, not necessarily the start, to be the summary. E.g.:

*For more background, see [this page](...).*

## Overview


This is the text I'd like in the summary, it's brief and to the point.


The rest of the article...