Domain Adaptive Object Detection (DAOD) generalizes the object detector from an annotated domain to a label-free novel one. Recent works estimate prototypes (class centers) and minimize the corresponding distances to adapt the cross-domain class conditional distribution. However, this prototype-based paradigm 1) fails to capture the class variance with agnostic structural dependencies, and 2) ignores the domain-mismatched classes with a sub-optimal adaptation. To address these two challenges, we propose an improved SemantIc-complete Graph MAtching framework, dubbed SIGMA++, for DAOD, completing mismatched semantics and reformulating adaptation with hypergraph matching. Specifically, we propose a Hypergraphical Semantic Completion (HSC) module to generate hallucination graph nodes in mismatched classes. HSC builds a cross-image hypergraph to model class conditional distribution with high-order dependencies and learns a graph-guided memory bank to generate missing semantics. After representing the source and target batch with hypergraphs, we reformulate domain adaptation with a hypergraph matching problem, i.e., discovering well-matched nodes with homogeneous semantics to reduce the domain gap, which is solved with a Bipartite Hypergraph Matching (BHM) module. Graph nodes are used to estimate semantic-aware affinity, while edges serve as high-order structural constraints in a structure-aware matching loss, achieving fine-grained adaptation with hypergraph matching. The applicability of various object detectors verifies the generalization of SIGMA++, and extensive experiments on nine benchmarks show its state-of-the-art performance on both AP 50 and adaptation gains.