Automated fact-checking (AFC) still falters on claims that are time-sensitive, entity-ambiguous, or buried beneath noisy search-engine results. We present PASS-FC, a Progressive and Adaptive Search Scheme for Fact Checking. Each atomic claim is first grounded with a precise time span and disambiguated entity descriptors. An adaptive search loop then issues structured queries, filters domains through credible-source selection, and expands queries cross-lingually; when necessary, a lightweight reflection routine restarts the loop. Experiments on six benchmark--covering general knowledge, scientific literature, real-world events, and ten languages--show that PASS-FC consistently outperforms prior systems, even those powered by larger backbone LLMs. On the multilingual X-FACT set, performance of different languages partially correlates with typological closeness to English, and forcing the model to reason in low-resource languages degrades accuracy. Ablations highlight the importance of temporal grounding and the adaptive search scheme, while detailed analysis shows that cross-lingual retrieval contributes genuinely new evidence. Code and full results will be released to facilitate further research.