You mention the document is accessible, so I assume it is fully and correctly tagged. Have you checked in Acrobat Pro to make sure the Tab Order property of each page is set to Use Document Structure? You can check this in the Page Thumbnails pane by selecting all the pages then right-click > properties (at least on a PC - I assume Mac works the same). If so the tab order and reading order should both follow the tag structure.
Hope this helps.
a 'C' student