Scaling inference for large language models is increasingly constrained by limited GPU memory, primarily due to the expanding intermediate states (KV caches) required for long-context generation and multi-user workloads. Once the KV cache exceeds the capacity of high-bandwidth memory, it must be offloaded to host memory and reloaded on demand, a workflow severely bottlenecked by the CPU–GPU interconnect, typically PCIe. Existing approaches exploiting offload KV caches to CPU memory and selectively reload partial segments for attention computation often underutilize CPU compute resources and suffer from accuracy degradation. We present Beyond, a drop-in runtime that integrates a smart offloading scheme to selectively identify and retain salient KV entries across continuous decoding sessions, together with a hybrid CPU–GPU attention mechanism for scalable inference. Beyond executes dense attention over recent KV entries stored in GPU memory while performing parallel, per-head sparse attention on salient contextual KV entries residing in CPU memory. The outputs are fused efficiently through a log-sum-exp scheme. During the bandwidth-constrained decoding phase, oversized KV caches are processed cooperatively by the aggregated CPU and GPU memory bandwidth, with only minimal PCIe data movement. Experiments across diverse models and workloads demonstrate that Beyond improves scalability, supports longer sequences and larger batch sizes, and outperforms existing sparse attention baselines in both efficiency and accuracy—all on commodity GPU hardware.