Individual-level mobility prediction is central to urban simulation, transportation planning, and policy analysis. Supervised sequence models achieve strong accuracy but require task-specific training and offer limited decision-level transparency. Recent LLM-based methods improve interpretability, yet mostly rely on static prompts and single-pass inference, limiting their ability to seek additional evidence when mobility signals are weak or conflicting. We propose \method{}, a training-free LLM-driven agent framework that formulates next-location prediction as adaptive evidence-controlled decision making. \method{} resolves routine cases through a fast path based on historical regularity, while ambiguous cases trigger iterative tool use over recent trajectories, historical behavior, stay-move likelihood, and geographical evidence. Across three mobility datasets, AgentMob achieves the strongest overall performance among training-free LLM-based methods, with GPT-5.4 reaching 71.42\% Acc@1 on BW, 33.14\% on YJMob100K, and 33.50\% on Shanghai ISP. On BW non-fast-path cases, the LLM controller improves Acc@1 from 30.65\% to 48.62\% over a same-tool statistical baseline, showing that its main benefit lies in resolving ambiguous predictions through adaptive evidence gathering. Our code is available at https://github.com/Unknown-zoo/AgentMob.