Hyperspectral image (HSI) reconstruction is vital for recovering spatial-spectral information from compressed measurements in coded aperture snapshot spectral imaging (CASSI) systems. Despite the effectiveness of end-to-end and deep unfolding methods, their reliance on substantial training data poses challenges, notably the scarcity of labeled HSIs. Existing approaches often train on limited datasets, such as KAIST and CAVE, leading to biased models with poor generalization capabilities. Addressing these challenges, we propose a universal Self-Supervised Adapter for Hyperspectral Snapshot Compressive Imaging (SAH-SCI). Unlike full fine-tuning or linear probing, SAH-SCI enhances model generalization by training a lightweight adapter while preserving the original model's parameters. We propose a novel approach that combines spectral and spatial adaptation to enhance an image model's capacity for spatial-spectral reasoning. Additionally, we introduce a customized adapter self-supervised loss function that captures the consistency, group invariance and image uncertainty of CASSI imaging. This approach effectively reduces the solution space for ill-posed HSI reconstruction. Experimental results demonstrate SAH's superiority over previous methods with fewer parameters, offering simplicity and adaptability to any end-to-end or unfolding methods. Our approach paves the way for leveraging more robust image foundation models in future hyperspectral imaging tasks.