My classic-jekyll-theme does something like this. When disabling the drop-down menu it will create a vertical menu that displays the selected main menu (book title?), with sub items (chapters?) and subsub items (Sections?) in it.
https://balancingrock.github.io/classic-jekyll-theme/
Though the example site at the link has the drop-down enabled, thus does not show the vertical menu.
Even when you don’t like the theme, I think it can easily be used as an example that you can modify yourself in order to create a special book-content widget.