Though big progress in table-to-text works, effectively leveraging table structure signals, e.g., hierarchical structure, remains challenging. Besides, deliberating generated descriptions proves to be effective for table-to-text. However, determining the appropriate outcome when encountering multi-pass candidates is another challenge. To this end, we propose a novel table-to-text approach on top of Self-evaluated multi-pass Generation and Heterogenous Multidominance Attention, namely SG-HMA. Specifically, we formulate the table structure into a multidominance (MD) structure and devise a heterogenous multidominance attention (HMA) to comprehensively explore the complex interactions encoded in the hierarchical structure, which can further deliver rich signals for text generation with the help of pre-trained language models (PLMs). Afterward, a contrastive loss is introduced to align the generation objective with evaluation metrics, so the more faithful generated descriptions can be guaranteed. We conduct extensive experiments on three public datasets, demonstrating that SG-HMA outperforms several SOTA methods quantitatively and qualitatively.