Magnetically hidden order is a hypernym for electronic ordering phenomena that are visible to macroscopic thermodynamic probes but whose microscopic symmetry cannot be revealed with conventional neutron or x-ray diffraction. In a handful off-electron systems, the ordering of odd-rank multipoles leads to order parameters with a vanishing neutron cross section. Among them, Ce3Pd20Si6 is known for its unique phase diagram exhibiting two distinct multipolar-ordered ground states (phases II and II'), separated by a field-driven quantum phase transition associated with a putative change in the ordered quadrupolar moment from O-2(0) to O-xy. Using torque magnetometry at sub-kelvin temperatures, here we find another phase transition at higher fields above 12 T, which appears only for low-symmetry magnetic field directions B parallel to (11L) with 1 < L ]. We also find from inelastic neutron scattering that the number of nondegenerate collective excitations induced by the magnetic field correlates with the number of phases in the magnetic phase diagram for the same field direction. Furthermore, the magnetic excitation spectrum suggests that the new phase II '' may have a different propagation vector, revealed by the minimum in the dispersion that may represent the Goldstone mode of this hidden-order phase.