Current techniques for Out-of-Distribution (OoD) detection predominantly rely on quantifying predictive uncertainty and incorporating model regularization during the training phase, using either real or synthetic OoD samples. However, methods that utilize real OoD samples lack exploration and are prone to overfit the OoD samples at hand. Whereas synthetic samples are often generated based on features extracted from training data, rendering them less effective when the training and OoD data are highly overlapped in the feature space. In this work, we propose a Wasserstein-score-based generative adversarial training scheme to enhance OoD detection accuracy, which, for the first time, performs data augmentation and exploration simultaneously under the supervision of limited OoD samples. Specifically, the generator explores OoD spaces and generates synthetic OoD samples using feedback from the discriminator, while the discriminator exploits both the observed and synthesized samples for OoD detection using a predefined Wasserstein score. We provide theoretical guarantees that the optimal solutions of our generative scheme are statistically achievable through adversarial training in empirical settings. We then demonstrate that the proposed method outperforms state-of-the-art techniques on various computer vision datasets and exhibits superior generalizability to unseen OoD data.