In this paper, the growths of ε-Ga₂O₃ thin films on c-plane sapphire substrate by metal-organic chemical vapor deposition (MOCVD) via single-step and two-step growth methods were studied respectively. High-resolution X-ray single crystal diffraction was used to analyze the phase composition of Ga₂O₃ along the c-axis direction of the sapphire. In the single-step method, the thin film grows directly on the sapphire substrate. The β-Ga₂O₃ ($\bar{4}$02) peak can be observed at growth temperatures ranging from 360 ℃ to 425 ℃, while the ε-Ga₂O₃ (004) peak can also be detected at 370~410 ℃. Samples grown at 380 and 390 ℃ exhibit stronger ε-Ga₂O₃ (004) peaks with lower full width at half maximum (FWHM) and a surface roughness. The two-step growth method involves using an ε-Ga₂O₃ thin film grown at 380 ℃ as a buffer layer, followed by continuing the growth of ε-Ga₂O₃ thin films at 400~430 ℃. It is observed that the intensity of the ε-Ga₂O₃ (004) peak is higher than that of the single-step growth method, with even lower FWHM values. The rocking curve FWHM of the (004) peak of the film reaches 0.49° at continuing growth temperature of 430 ℃. Further adjustment of the pressure in the single-step growth method confirms that the buffer layer effectively promotes the growth of ε-Ga₂O₃ along the c-axis.