Given a list of points on a sphere find the place for another point such that most space is covered?
The general problem is called Circle packing, which gets easier with equiradius circles, to get hexagonal grid and put circles on hexagon vertices and one in the center. This works on Euclidian space, so the hexagon coordinates should be converted by some kind of mapping, for example Vincenty formulae.

This method will give really neglible overlapping, but to be exact further corrections might be needed (like error calculation at intersection points to shift the centers). For small radius (like 50m) error will be very small (using the mapping to sphere). Performing grow on already available circles will give the problem of initial rotation to fit more space, so with many initial circles not fitting the grid only general method (iterative one) will give the optimum.

The deterministic optimization shows the method to treat circle packing as quadratic optimization.