Moment methods, which are powerful and simple techniques for analyzing the reliability of a system, evaluate the statistical moments of a system response function and use information from the probability distribution in the analysis. The full factorial moment method (FFMM) performs reliability analysis by using a $3^n$ full factorial design of experiments (DOE) and the Pearson system for random variables. To overcome the inefficiency of FFMM, the response surface moment method (RSMM) has been proposed, which is based on a response surface model (RSM) that is updated by adding cross product terms into the simple quadratic model. In this paper, we propose the enhanced RSMM (RSMM+) that modifies the procedure of selecting a cross product term in the RSMM and adds a process of judging whether the response surface model can be established before performing an additional experiment. We apply the proposed method to several examples and show that it gives better results in efficiency.