In-context reinforcement learning (ICRL) promises fast adaptation to unseen environments without parameter updates, but current methods either cannot improve beyond the training distribution or require near-optimal data, limiting practical adoption. We introduce SPICE, a Bayesian ICRL method that learns a prior over Q-values via deep ensemble and updates this prior at test-time using in-context information through Bayesian updates. To recover from poor priors resulting from training on sub-optimal data, our online inference follows an Upper-Confidence Bound rule that favours exploration and adaptation. We prove that SPICE achieves regret-optimal behaviour in both stochastic bandits and finite-horizon MDPs, even when pretrained only on suboptimal trajectories. We validate these findings empirically across bandit and control benchmarks. SPICE achieves near-optimal decisions on unseen tasks, substantially reduces regret compared to prior ICRL and meta-RL approaches while rapidly adapting to unseen tasks and remaining robust under distribution shift.