Reliability analysis is time consuming, and high efficiency could be maintained through the integration of the Kriging method and Monte Carlo simulation (MCS). This Kriging-based MCS reduces the computational cost by building a surrogate model to replace the original limit-state function through MCS. The objective of this research is to further improve the efficiency of reliability analysis with a new strategy for building the surrogate model. The major approach used in this research is to refine (update) the surrogate model by accounting for the full information available from the Kriging method. The existing Kriging-based MCS uses only partial information. Higher efficiency is achieved by the following strategies: (1) a new formulation defined by the expectation of the probability of failure at all the MCS sample points, (2) the use of a new learning function to choose training points (TPs). The learning function accounts for dependencies between Kriging predictions at all the MCS samples, thereby resulting in more effective TPs, and (3) the employment of a new convergence criterion. The new method is suitable for highly nonlinear limit-state functions for which the traditional first- and second-order reliability methods (FORM and SORM) are not accurate. Its performance is compared with that of existing Kriging-based MCS method through five examples.