You need to add the basemap to the basemap group within the organization. Log in as your admin. Create a new map.
Add a layer from the web and make sure the Use as basemap is checked.
Save the map and share in the Basemap group.
Then it should show up in the basemap group.
From there it should be in the basemap gallery.
Then users can select an esri basemap and come back to your custom one.