In several metals, including URhGe, superconductivity has recently been observed to appear and coexist with ferromagnetism at temperatures well below that at which the ferromagnetic state forms. However, the material characteristics leading to such a state of coexistence have not yet been fully elucidated. We report that in URhGe there is a magnetic transition where the direction of the spin axis changes when a magnetic field of 12 tesla is applied parallel to the crystal b axis. We also report that a second pocket of superconductivity occurs at low temperature for a range of fields enveloping this magnetic transition, well above the field of 2 tesla at which superconductivity is first destroyed. Our findings strongly suggest that excitations in which the spins rotate stimulate superconductivity in the neighborhood of a quantum phase transition under high magnetic field.