The application of high-precision value-flow analysis is experiencing a paradigm shift from planned executions to online ad hoc queries driven by human auditors and AI agents. However, existing techniques struggle in this interactive setting: exhaustive offline tabulation is fundamentally intractable, while memoryless online search suffers from redundant exploration and SMT invocations. To bridge this gap, we propose Sponge, a novel two-phase framework that accelerates ad hoc queries through boundary-anchored indexing. Offline, Sponge employs an adaptive-depth strategy to selectively precompute feasible value-flow segments at critical procedure boundaries, optimizing SMT allocation based on traversal probability and search space complexity. Online, it utilizes an index-guided push-down search with lazy expansion to dynamically stitch these pre-verified segments, effectively bypassing redundant state exploration and pruning unsatisfiable paths. We evaluated Sponge on 9 C/C++ projects (up to 3.8M LoC). Results demonstrate that Sponge drops the 95th-percentile online query time from nearly 270s to under 50s compared to a baseline search. Furthermore, the adaptive strategy reduces offline indexing time by 75% over a uniform approach, amortizing the offline cost in fewer than 300 queries for workloads dominated by complex queries.