  Precise characterization of a system's Hamiltonian is crucial to its
high-fidelity control that would enable many quantum technologies, ranging from
quantum computation to communication and sensing. In particular, non-secular
parts of the Hamiltonian are usually more difficult to characterize, even if
they can give rise to subtle but non-negligible effects. Here we present a
strategy for the precise estimation of the transverse hyperfine coupling
between an electronic and a nuclear spin, exploiting effects due to forbidden
transitions during the Rabi driving of the nuclear spin. We applied the method
to precisely determine the transverse coupling between a Nitrogen-Vacancy
center electronic spin and its Nitrogen nuclear spin. In addition, we show how
this transverse hyperfine, that has been often neglected in experiments, is
crucial to achieving large enhancements of the nuclear Rabi driving.
