The local electronic and magnetic properties of superconducting FeSe have been investigated by Kβ x-ray emission and simultaneous x-ray absorption spectroscopy (XAS) at the Fe K edge at high pressure and low temperature. Our results indicate a sluggish decrease of the local Fe spin moment under pressure up to 7 GPa, in line with previous reports, followed by a sudden increase at higher pressure. The magnetic surge is preceded by an abrupt change of the Fe local structure as observed by the decrease of the XAS preedge region intensity and corroborated by ab initio simulations. This pressure corresponds to a structural transition from the Cmma form to the denser P bnm form with octahedral coordination of iron. Finally, the near-edge region of the XAS spectra shows a change before this transition at 5 GPa, corresponding well with the onset pressure of the sudden enhancement of T c. Our results emphasize the delicate interplay between structural, magnetic, and superconducting properties in FeSe under pressure.