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Stacking symmetry of 2D magnet CrI₃ plays a key role in defining the magnetic ground states. It has been shown that hydrostatic pressure can switch the layered AFM to FM state via tuning the layer stacking from monoclinic to rhombohedral in bilayer CrI₃. A natural question would be: does crystal structure also respond to the magnetic phase transition in N-layer CrI₃? In this work, we investigate layer-number, temperature, and magnetic field dependence of Raman spectra in few-layer CrI₃ using polarization-resolved Raman spectroscopy. Instead of focusing on an A_g phonon mode at about 129 cm^-1, we focus on a doublet degenerate E_g phonon mode at ~107 cm^-1. A monoclinic stacking induced degeneracy lift from E_g in monolayer to A_g+B_g in N-layer (N = 2, 3, 4, …) CrI₃ is demonstrated in layer-number dependent results. Such a monoclinic structure exhibits a failed attempt to transit into the rhombohedral structure upon cooling. This monoclinic structure undergoes a further monoclinic distortion across the magnetic field-induced layered AFM-to-FM transition. Our results show that the crystal structure does respond to the magnetic phase transition in N-layer CrI₃.