We propose that noncollinear magnetic order in magnetic systems can harbor a higher-order topological magnon phase with non-Hermitian topology and hinge magnon modes. We consider a three-dimensional system of interacting local moments on stacked layers of honeycomb lattice. It initially favors a collinear magnetic order along an in-plane direction, which turns into a noncollinear order upon applying an external magnetic field perpendicular to the easy axis. We exploit the non-Hermitian nature of the magnon Hamiltonian to show that this field-induced transition corresponds to the transformation from a topological magnon insulator to a higher-order topological magnon state with a one-dimensional hinge mode. As a concrete example, we discuss the recently discovered monoclinic phase of thin chromium trihalides, which we propose as a promising material candidate of the higher-order topological magnon phase.