To create custom mapserver or fix map that don't work anymore (by adding it new) you can read short tutorial here: https://codesector.kayako.com/article/6-setting-up-custom-map-servers
You can add custom maps and custom overlays at the same way.
For maps add the mapserver.txt to your android /maverick/tiles/your custom map/
For overlay add the mapserver.txt to /maverick/overlays/your custom overlay/
Here you find some working templates for maps and overlays for testing purposes.
Disclaimer:
Persisting map tiles locally breaks the terms of service for many popular basemap providers like Google.
If you start storing tiles offline without permission, then you do so at your own risk.
It is highly recommend to consult and follow usage policies of a given server!
Respect their terms of use/tile usage policy.
additional source: https://wiki.openstreetmap.org/wiki/Tiles