For power-limited mobile devices, how to prolong battery life is an important issue. The voice communication needs to implement high computation complexity echo cancellation to obtain a satisfied voice quality. Partial-update-based adaptive algorithms are known solutions to reduce the computation complexity for long-tap adaptation, such as echo cancellation. However, the convergence rate is decreased due to the partial update algorithm only updates part of weights of the adaptive filters. In this paper, we propose an enhanced switching-based variable step-size approach for the M-max partial update LMS algorithms. During the initial stage, the errors are dominated by the mismatch between the tap weights of the adaptive filter and its ideal weights. We correlate the squared error signals with polynomial of error signal to gain a large step-size; on the other hand, during the steady state, the errors mainly come from the additive noise. Therefore, we switch the correlation to the other mode so that the effect of noise can be eliminated. This can be done by correlating the error signals with a delayed version of error signals and hence a small step-size is obtained during the late stages. Simulation results show that when only one fourth of all taps are updated in one iteration, the proposed method significantly enhances the convergence rate of the M-max least-mean-square (LMS) algorithms.