  This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on
the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann
surfaces of any genus. This generalizes the existence results by C. LeBrun in
arXiv:1411.3992 and arXiv:1504.06669. Moreover, by calculating the (normalized)
Einstein-Hilbert functional of our examples we generalize Theorem E of
arXiv:1504.06669, which speaks to the abundance of Hermitian Einstein-Maxwell
solutions on such manifolds. As a bonus, we exhibit certain pairs of strongly
Hermitian Einstein-Maxwell solutions, first found in arXiv:1504.06669, on the
first Hirzebruch surface in a form which clearly shows that they are conformal
to a common K\"ahler metric. In particular, this yields a non-trivial example
of non-uniqueness of positive constant scalar curvature metrics in a given
conformal class.
