Optimizing molecules under small evaluation budgets requires deciding both where in chemical space to search and which candidate within that region to evaluate. We introduce Trust Region Adaptive Chemical Exploration (TRACE), a framework for open-ended molecular Bayesian optimization (BO), where candidates are generated on demand rather than selected from a fixed library. TRACE combines an objective-free graph-perturbation mechanism with a fixed, chemistry-aware Gaussian process (GP) surrogate using fragprints, a representation combining molecular fingerprints and fragment descriptors. Rather than learning the search geometry itself, TRACE makes the scale of chemical exploration explicit through a Tanimoto-similarity threshold controlling chemical extent and a cardinality budget controlling candidate-search breadth. A simple success/failure controller expands or contracts both quantities during optimization, while the GP and acquisition function select which admissible molecules to evaluate. Across six challenging molecular optimization benchmarks under matched 1,000-evaluation budgets, TRACE achieves the strongest overall sample efficiency as measured by the best molecule found as evaluations accumulate. Crossed ablations on two objectives further show that adaptive region control contributes beyond the proposal mechanism, while its benefit depends on the proposal distribution. We further apply TRACE to the stability of iron phthalocyanine (FePc), a non-precious molecular electrocatalyst whose degradation can involve loss of the Fe center. Using a first-principles density functional theory (DFT) workflow designed for accurate energy evaluation, TRACE increases the computed thermodynamic resistance to Fe dissolution from 0.42 to 2.59 eV over 80 designed evaluations.

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