Smart contracts manage substantial financial value yet resist testing: valid inputs follow an application binary interface (ABI), exploitable bugs need multi-step stateful sequences, and each Ethereum Virtual Machine (EVM) execution is costly, so executions bind. We present SSCFuzz, which separates \emph{what to attack} from \emph{how}: a selector picks a strategy and a generate-or-mutate mode each iteration, a large language model (LLM) creates the transaction sequence from the contract's source, and Foundry executes it against the real EVM, supplying bytecode coverage and a balance-invariant financial-loss oracle. Because a cold-start budget affords about eleven trials per strategy, the selector is a training-free reinforcement learner: it estimates each arm's value online from its own reward, but \emph{encodes} the non-stationary switching rule (warm up, exploit the best recent-payoff arm, abandon it once it stops paying) rather than learning it. Against four baselines on 33 SmartBugs-Curated contracts and 25 real-world DeFiHackLabs exploit reproductions, SSCFuzz detects exploits in 48\% of contracts on both corpora, leading every baseline on the real-world one (best 40\%) and tying the best on the academic one. Further analysis reveals complementary effects. On complex contracts, LLM generation achieves 0.114 lower coverage than uniform ABI sampling ($\delta=-0.50$) but increases exploit detection by 3.5$\times$, showing that coverage alone does not characterize exploit-finding performance. Uniform-ABI exploration uses only one eighth of the iterations yet contributes half of new coverage and the first exploit for roughly half of the solved contracts.