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Hyperspectral image (HSI) restoration is crucial for reliable analysis, as real HSIs suffer from degradations like noise, blur, and resolution loss. However, existing models trained on source data often fail on target domains lacking clean references, a common occurrence in practice. To address this issue, we present HIR-ALIGN, a plug-and-play target-adaptive augmentation framework that enhances hyperspectral image restoration by augmenting limited training images with synthetic data that closely matches the target distribution using no extra data. It consists of three stages: (i) proxy generation, where off-the-shelf restoration models restore degraded target observations to produce semantics-preserving proxy HSIs that approximate target-domain clean images; (ii) distribution-adaptive synthesis, where a blur-robust unCLIP diffusion model generates target-aligned RGBs from proxy RGBs, with prompt conditioning and embedding-space noise initialization. Then, a warp-based spectral transfer module synthesizes HSIs by aligning each generated RGB with the proxy RGB, estimating soft patch-wise transport weights, and applying these weights and learnable local interpolation kernels to the proxy HSI; and (iii) aligned supervised finetuning, where restoration networks pretrained on the source distribution are finetuned using both the proxy HSIs and synthesized target-aligned HSIs, and are then deployed on degraded target images. We further provide theoretical analysis showing that augmentation-based finetuning can achieve lower target-domain restoration risk by jointly improving target distribution coverage and controlling spectral bias. Extensive experiments on simulated and real datasets across denoising and super-resolution tasks demonstrate that HIR-ALIGN consistently improves source-only supervised baselines, outperforming both source-only counterparts and representative unsupervised methods.