You will need to add a new reading list content item for each section. Do this by hovering over the module, click the +, select 'Content market' then the + icon in the option marked 'Reading list'. Click on this newly-created content item to open the Talis Aspire overlay, find the correct reading list if not displayed by default, then choose appropriate section from the drop-down box. With the right section in the preview, click the 'save' button to confirm then the X in the top-left to exit. Set the link as visible to students and try the student preview to confirm. NB: You can also add links to reading lists or sections within them whenever there is the option to insert an 'LTI Item'. The process is the same as described above. This video
depicts the process (the video has no sound).