Phase noise is a clue performance indicator for MEMS-based resonant sensors. The optimal resolution achievable with these sensors is limited by the close-to-the-carrier phase noise resulting from the modulation of noise sources by the mechanical resonator. In this paper, we focus on the effect of a white noise input source on a one-sided capacitive MEMS resonator. We study how the closed-loop phase shift affects its frequency stability. Our study reveals the existence of optimal points, with phase noise minimization and SNR maximization, which cannot be predicted by a traditional third-order Taylor-Series approach. We reveal that, due to the modulation of the actuation force by the electrostatic nonlinearity, tuning the closed-loop phase shift to improve the oscillator frequency stability is especially relevant.