Customer expansion, i.e., growing a business existing customer base by acquiring new customers, is critical for scaling operations and sustaining the long-term profitability of logistics companies. Although state-of-the-art works model this task as a single-node classification problem under a heterogeneous graph learning framework and achieve good performance, they struggle with extremely positive label sparsity issues in our scenario. Multi-task learning (MTL) offers a promising solution by introducing a correlated, label-rich task to enhance the label-sparse task prediction through knowledge sharing. However, existing MTL methods result in performance degradation because they fail to discriminate task-shared and task-specific structural patterns across tasks. This issue arises from their limited consideration of the inherently complex structure learning process of heterogeneous graph neural networks, which involves the multi-layer aggregation of multi-type relations. To address the challenge, we propose a Structure-Aware Hierarchical Information Sharing Framework (SrucHIS), which explicitly regulates structural information sharing across tasks in logistics customer expansion. SrucHIS breaks down the structure learning phase into multiple stages and introduces sharing mechanisms at each stage, effectively mitigating the influence of task-specific structural patterns during each stage. We evaluate StrucHIS on both private and public datasets, achieving a 51.41% average precision improvement on the private dataset and a 10.52% macro F1 gain on the public dataset. StrucHIS is further deployed at one of the largest logistics companies in China and demonstrates a 41.67% improvement in the success contract-signing rate over existing strategies, generating over 453K new orders within just two months.