Phase manipulation is essential to quantum information processing, for which the orbital angular momentum (OAM) of photon is a promising high-dimensional resource. Dove prism (DP) is one of the most important elements to realize the nondestructive phase manipulation of OAM photons. DP usually changes the polarization of light and thus increases the manipulation error for a spin-OAM hybrid state. DP in a Sagnac interferometer also introduces a mode-dependent global phase to the OAM mode. In this work, we implemented a high-dimensional controlled-phase manipulation module (PMM), which can compensate the mode-dependent global phase and thus preserve the phase in the spin-OAM hybrid superposition state. The PMM is stable for free running and is suitable to realize the high-dimensional controlled-phase gate for spin-OAM hybrid states. Considering the Sagnac-based structure, the PMM is also suitable for classical communication with the spin-OAM hybrid light field.