The advancement of Time Series Foundation Models (TSFMs) has been driven primarily by large-scale pre-training, but inference-time compute potential remains largely untapped. This work systematically investigates two questions: how do TSFMs behave under standard sampling-based inference scaling, and can controlled sampling diversity enhance performance? We first examine the properties of TSFMs under standard sampling often fail to adhere to scaling laws due to insufficient exploration of the solution space. Building on this, we then delve into diversified inference scaling via tailored time series perturbations to expand the generative distribution's support. We theoretically analyze the diversity-fidelity trade-off and derive a critical sample threshold for diversified sampling to outperform standard sampling. Extensive experiments across various TSFMs and datasets show proper diversified inference scaling yields substantial performance gains without parameter updates, establishing inference design as a critical, compute-efficient dimension of TSFM optimization. As an application, we propose RobustMSE, a rigorous metric to quantify the headroom performance of TSFM under a fixed budget. Overall, our findings clarify these factor interactions, enabling reliable performance via diverse large-scale inference time series in parallel environments without re-training TSFMs.