Added code so the plugin would create a new custom taxonomy term matching the story name every time a story is created. Then it automatically adds the taxonomy term to each new post (chapter) that is published which also contains that story in its meta data. This associates new chapters with a specific story, allowing readers to subscribe to new posts with that taxonomy term (see plugin edit 2).
Set the plugin to use the custom taxonomy I created for stories (see plugin edit 1) instead of categories. Then I modified the loop that would build a list of possible taxonomy terms to subscribe to. I made it find the story name of the currently viewed post and only display that taxonomy term as an option in the widget. This way, a reader will only see an option to subscribe to the story they are currently reading.