We present a novel architecture aimed towards incremental construction and exploitation of a hierarchical 3D scene graph representation during semantic-aware inspection missions. Inspection planning, particularly of distributed targets in previously unseen environments, presents an opportunity to exploit the semantic structure of the scene during reasoning, navigation and scene understanding. Motivated by this, we propose the 3D Layered Semantic Graph (3DLSG), a hierarchical inspection scene graph constructed in an incremental manner and organized into abstraction layers that support planning demands in real-time. To address the task of semantic-aware inspection, a mission framework, termed as Enhanced First-Look Inspect Explore (xFLIE), that tightly couples the 3DLSG with an inspection planner is proposed. We assess the performance through simulations and experimental trials, evaluating target-selection, path-planning and semantic navigation tasks over the 3DLSG model. The scenarios presented are diverse, ranging from city-scale distributed to solitary infrastructure targets in simulated worlds and subsequent outdoor and subterranean environment deployments onboard a quadrupedal robot. The proposed method successfully demonstrates incremental construction and planning over the 3DLSG representation to meet the objectives of the missions. Furthermore, the framework demonstrates successful semantic navigation tasks over the structured interface at the end of the inspection missions. Finally, we report multiple orders of magnitude reduction in path-planning time compared to conventional volumetric-map-based methods over various environment scale, demonstrating the planning efficiency and scalability of the proposed approach.