Autonomous long-horizon mobile manipulation encompasses a multitude of challenges, including scene dynamics, unexplored areas, and error recovery. Recent works have leveraged foundation models for scene-level robotic reasoning and planning. However, the performance of these methods degrades when dealing with a large number of objects and large-scale environments. To address these limitations, we propose MORE, a novel approach for enhancing the capabilities of language models to solve zero-shot mobile manipulation planning for rearrangement tasks. MORE leverages scene graphs to represent environments, incorporates instance differentiation, and introduces an active filtering scheme that extracts task-relevant subgraphs of object and region instances. These steps yield a bounded planning problem, effectively mitigating hallucinations and improving reliability. Additionally, we introduce several enhancements that enable planning across both indoor and outdoor environments. We evaluate MORE on 81 diverse rearrangement tasks from the BEHAVIOR-1K benchmark, where it becomes the first approach to successfully solve a significant share of the benchmark, outperforming recent foundation model-based approaches. Furthermore, we demonstrate the capabilities of our approach in several complex real-world tasks, mimicking everyday activities. We make the code publicly available at https://more-model.cs.uni-freiburg.de.