We introduce ASAP2, a variant of the batch-means algorithm ASAP for steady-state simulation output analysis. ASAP2 operates as follows: the batch size is progressively increased until the batch means pass the Shapiro-Wilk test for multivariate normality; and then ASAP delivers an adjusted confidence interval. The latter adjustment is based on an inverted Cornish-Fisher expansion for the classical NOBM t-ratio, where the terms of the expansion are estimated via an order-one autoregressive time series model of the batch means. In comparison to the widely used ABATCH and LBATCH algorithms, we found ASAP2 to deliver confidence intervals that not only satisfy a user-specified absolute or relative precision requirement but also outperform ABATCH and LBATCH with respect to confidence-interval coverage probability.