Interpreting natural-language commands to localize target objects is critical for autonomous driving (AD). Existing visual grounding (VG) methods for autonomous vehicles (AVs) typically struggle with ambiguous, context-dependent instructions, as they lack reasoning over 3D spatial relations and anticipated scene evolution. Grounded in the principles of world models, we propose ThinkDeeper, a framework that reasons about future spatial states before making grounding decisions. At its core is a Spatial-Aware World Model (SA-WM) that learns to reason ahead by distilling the current scene into a command-aware latent state and rolling out a sequence of future latent states, providing forward-looking cues for disambiguation. Complementing this, a hypergraph-guided decoder then hierarchically fuses these states with the multimodal input, capturing higher-order spatial dependencies for robust localization. In addition, we present DrivePilot, a multi-source VG dataset in AD, featuring semantic annotations generated by a Retrieval-Augmented Generation (RAG) and Chain-of-Thought (CoT)-prompted LLM pipeline. Extensive evaluations on six benchmarks, ThinkDeeper ranks #1 on the Talk2Car leaderboard and surpasses state-of-the-art baselines on DrivePilot, MoCAD, and RefCOCO/+/g benchmarks. Notably, it shows strong robustness and efficiency in challenging scenes (long-text, multi-agent, ambiguity) and retains superior performance even when trained on 50% of the data.