  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 of f-electron systems, the ordering of odd-rank multipoles leads to
order parameters with a vanishing neutron cross-section. Among them,
Ce$_3$Pd$_{20}$Si$_6$ 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 subkelvin temperatures, here we find another phase transition at higher
fields above 12 T, which appears only for low-symmetry magnetic field
directions $\mathbf{B} \parallel \langle11L\rangle$ with $1 < L \leq 2$. While
the order parameter of this new phase II'' remains unknown, the discovery
renders Ce$_3$Pd$_{20}$Si$_6$ a unique material with two field-driven phase
transitions between distinct multipolar phases. They are both clearly
manifested in the magnetic-field dependence of the field-induced (111) Bragg
intensities measured with neutron scattering for $\mathbf{B} \parallel [112]$.
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.
