Compressive Sensing (CS) allows for the sampling of signals at well below the Nyquist rate but does so, usually, at the cost of the suppression of lower amplitude signal components. Recent work suggests that important information essential for recognizing targets in the radar context is contained in the side-lobes as well, which are often suppressed by CS. In this paper we extend existing techniques and introduce new techniques both for improving the accuracy of CS reconstructions and for improving the separability of scenes reconstructed using CS. We investigate the Discrete Wavelet Transform (DWT), and show how the use of the DWT as a representation basis may improve the accuracy of reconstruction generally. Moreover, we introduce the concept of using multiple wavelet-based reconstructions of a scene, given only a single physical observation, to derive reconstructions that surpass even the best wavelet-based CS reconstructions. Lastly, we specifically consider the effect of the wavelet-based reconstruction on classification. This is done indirectly by comparing outputs of different algorithms using a variety of separability measures. We show that various wavelet-based CS reconstructions are substantially better than conventional CS approaches at inducing (or preserving) separability, and hence may be more useful in classification applications.