In this work, we introduce a new acquisition function for sequential sampling to efficiently quantify rare-event statistics of an input-to-response (ItR) system with given input probability and expensive function evaluations. Our acquisition is a generalization of the likelihood-weighted (LW) acquisition [Sapsis, T.P., Output-Weighted Optimal Sampling for Bayesian Regression and Rare Event Statistics Using Few Samples, Proc. R. Soc. A, 476(2234):20190834, 2020; Sapsis, T.P. and Blanchard, A., Optimal Criteria and Their Asymptotic Form for Data Selection in Data-Driven Reduced-Order Modelling with Gaussian Process Regression, Philos. Trans. R. Soc. A, 380(2229):20210197, 2022], which was initially designed for the same purpose and then extended to many other applications. The improvement in our acquisition comes from the generalized form with two additional parameters. By adjusting these parameters, one can effectively target and address two weaknesses of the original LW acquisition: (1) that the input space associated with rare-event responses is not sufficiently stressed in sampling; (2) that the surrogate model (generated from samples) may have a significant deviation from the true ItR function, especially for cases with complex ItR function and limited number of samples. In addition, we develop a critical procedure in Monte Carlo discrete optimization of the acquisition function, which achieves orders-of-magnitude acceleration compared to existing approaches for such a type of problem. The superior performance of our new acquisition to the original LW acquisition is demonstrated in a number of test cases, including some cases that were designed to show the effectiveness of the original LW acquisition. We finally apply our method to an engineering example to quantify the rare-event roll-motion statistics of a ship in a random sea.