Hyperspectral imaging typically produces huge data volume that demands enormous computational resources in terms of storage, computation and transmission, particularly when real-time processing is desired. In this paper, we study a lowcomplexity scheme for hyperspectral imaging completely bypassing high-complexity compression task. In this scheme, compressive hyperspectral data are acquired directly by a device similar to the single-pixel camera based on the principle of compressive sensing (CS). To decode the compressive data, we propose a flexible recovery strategy by taking advantage of the joint spatial-spectral correlation model of hyperspectral images. Moreover, a thorough investigation is analytically conducted on compressive hyperspectral data and we find that the compressive data still have strong spectral correlation. To make the recovery more accurate, an adaptive spectral band reordering algorithm is directly added to the compressive data before the reconstruction by making best use of spectral correlation. The real hyperspectral images are tested to demonstrate the feasibility and efficiency of the proposed algorithm. Experimental results indicate that the proposed recover algorithm can speed up the reconstruction process with reliable recovery quality.