We investigate the dynamics of the carrier-envelope-offset (CEO) frequency, f(CEO), controlled by a pump current on the self-referencing of an optical frequency comb generated from a diode-pumped solid-state laser at 1.56 μm. We observe a reversal point in the tuning of f(CEO) with the pump current. Between the low- and high-frequency region in the dynamic response of f(CEO) to pump current modulation, we observe a significant phase shift of ≈180 deg in the transfer function. As a result, it is impossible to stabilize f(CEO) at a pump current above the reversal point, although the free-running CEO beat at this point has a higher signal-to-noise ratio than underneath the reversal point at which the locking is straightforward. Our results indicate that a high signal-to-noise ratio and a low-noise CEO beat are not sufficient indicators for the feasibility of comb self-referencing in cases for which CEO dynamics play a dominant role.