@JessicaWeinberg - If I'm understanding you correctly, yes you can do that.
Anytime you add any type of map (web map, web scene, express map) to a story you have the opportunity to set the extent/viewpoint for the map. You are not limited to the extent at which the map or the basemap was saved.
While creating an express map, go to the options (gear) tab in the side panel, click the basemap to open the basemap picker, click Browse more maps, and select the web map with your organization's basemap. After changing the basemap (and before closing the map designer) navigate the map to whatever area you want to show and click Done. When the map is placed in the story, you'll see the area you specified.
Currently, you have to update the basemap for each express map you add to a story. However, we're looking at adding the ability to set the basemap as part of a theme so your basemap just comes in immediately for each new express map you create. (It looks like you've already voted for/commented on that idea. 🙂 )
Owen Evans
Lead Product Engineer | StoryMaps