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You would need a small (MapBasic) application to add additional tables/layers to the Base Maps element in the Open dropdown.
5 (!) years ago I created a tool called Favorite Maps that could create an additional dropdown on some of the tabs in MapInfo Pro which would give you quick access to some of your favorite maps which could be base maps or basically any table.
This tool could get a refresh and be used to add other base maps to the Base Maps element on the Open dropdown.
The source code for Favorite Maps can be found on Github.