Benkeser et al. demonstrate how adjustment for baseline covariates in randomized trials can meaningfully improve precision for a variety of outcome types. Their findings build on a long history, starting in 1932 with R.A. Fisher and including more recent endorsements by the U.S. Food and Drug Administration and the European Medicines Agency. Here, we address an important practical consideration: *how* to select the adjustment approach -- which variables and in which form -- to maximize precision, while maintaining Type-I error control. Balzer et al. previously proposed *Adaptive Prespecification* within TMLE to flexibly and automatically select, from a prespecified set, the approach that maximizes empirical efficiency in small trials (N$<$40). To avoid overfitting with few randomized units, selection was previously limited to working generalized linear models, adjusting for a single covariate. Now, we tailor Adaptive Prespecification to trials with many randomized units. Using $V$-fold cross-validation and the estimated influence curve-squared as the loss function, we select from an expanded set of candidates, including modern machine learning methods adjusting for multiple covariates. As assessed in simulations exploring a variety of data generating processes, our approach maintains Type-I error control (under the null) and offers substantial gains in precision -- equivalent to 20-43\% reductions in sample size for the same statistical power. When applied to real data from ACTG Study 175, we also see meaningful efficiency improvements overall and within subgroups.