Reinforcement learning (RL) for large language models (LLMs) remains expensive, particularly because the rollout is expensive. Decoupling rollout generation from policy optimization (e.g., leveraging a more efficient model to rollout) could enable substantial efficiency gains, yet doing so introduces a severe distribution mismatch that destabilizes learning. We propose Jackpot, a framework that leverages Optimal Budget Rejection Sampling (OBRS) to directly reduce the discrepancy between the rollout model and the evolving policy. Jackpot integrates a principled OBRS procedure, a unified training objective that jointly updates the policy and rollout models, and an efficient system implementation enabled by top-$k$ probability estimation and batch-level bias correction. Our theoretical analysis shows that OBRS consistently moves the rollout distribution closer to the target distribution under a controllable acceptance budget. Empirically, \sys substantially improves training stability compared to importance-sampling baselines, achieving performance comparable to on-policy RL when training Qwen3-8B-Base for up to 300 update steps of batchsize 64. Taken together, our results show that OBRS-based alignment brings us a step closer to practical and effective decoupling of rollout generation from policy optimization for RL for LLMs.