We propose a network-model realization of magnetic higher-order topological phases (HOTPs) in the presence of the combined space-time symmetry C4T - the product of a fourfold rotation and time-reversal symmetry. We show that the system possesses two types of HOTPs. The first type, analogous to Floquet topology, generates a total of eight corner modes at 0 or π eigenphase, while the second type, hidden behind a weak topological phase, yields a unique phase with eight corner modes at ±π/2 eigenphase (after gapping out the counterpropagating edge states), arising from the product of particle-hole and phase-rotation symmetry. By using a bulk Z4 topological index (Q), we found both HOTPs have Q=2, whereas Q=0 for the trivial and the conventional weak topological phase. Together with a Z2 topological index associated with the reflection matrix, we are able to fully distinguish all phases. Our work motivates further studies on magnetic topological phases and symmetry-protected 2π/n boundary modes, as well as suggesting that such phases may find their experimental realization in coupled-ring-resonator networks.