Win measures, including the win ratio (WR), win odds (WO), net benefit (NB), and desirability of outcome ranking (DOOR), are increasingly used in randomized clinical trials with multiple hierarchical ordinal endpoints. In practice, however, one or more component endpoints may have missing data. The standard pairwise-comparison approach, which treats pairs with missing outcomes as ties, can produce biased estimates, even if the data are missing completely at random (MCAR). Although inverse probability of censoring weighting (IPCW) methods have been developed for censored survival endpoints, corresponding methods for addressing missing hierarchical ordinal endpoints are not yet available. To address this gap, we develop inverse probability weighting (IPW) and augmented IPW (AIPW) estimators for win measures with hierarchical ordinal endpoints subject to missing data, allowing missingness to depend on treatment assignment and baseline covariates. The IPW estimator corrects bias by reweighting complete observed outcomes using joint non-missingness probabilities involved in estimating the joint cell probabilities that define the win measures. The AIPW estimator additionally incorporates outcome modeling, improving efficiency and achieving double robustness. For inference, we derive closed-form variance estimators for both methods based on influence functions. Simulation studies show that the standard approach can be substantially biased, whereas the proposed IPW and AIPW estimators remain consistent with near-nominal coverage. Furthermore, the AIPW estimator is generally more efficient than IPW estimator. Applications to the SCOUT-CAP and ACTT-1 trials illustrate the practical utility of the proposed methods. An R package, WinMO, is provided for implementation.