Existing hard-label text attacks often rely on inefficient "outside-in" strategies that traverse vast search spaces. We propose PivotAttack, a query-efficient "inside-out" framework. It employs a Multi-Armed Bandit algorithm to identify Pivot Sets-combinatorial token groups acting as prediction anchors-and strategically perturbs them to induce label flips. This approach captures inter-word dependencies and minimizes query costs. Extensive experiments across traditional models and Large Language Models demonstrate that PivotAttack consistently outperforms state-of-the-art baselines in both Attack Success Rate and query efficiency.