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…Some examples of base loaded antennas ground the bottom of the loading coil. The feed point is tapped at a point above the bottom of the coil where ideally the load can be matched to the transmitter. …
This could be worth a try, but probably there is no point along the length of a coil with one end connected to r-f ground where the reactance is zero, so the SWR at those points will be high.
Another consideration is that the loading coils are internal in many modern Part 15 AM transmitters, making it difficult to find a tap point by experiement. Also I think neither end of those coils connect to r-f ground.