For a multiband signal, the minimum sampling rate required for an arbitrary sampling method, which allows perfect reconstruction, is NB, where N is the number of bands and B is the maximum bandwidth. It has been proposed in the literature that, if the carrier frequency information of a multiband signal is not known apriori, then we require a minimum sampling rate of 2NB for perfect reconstruction. Modulated wideband converter (MWC) is a recently introduced blind sampling method. Unlike the traditional sampling methods, where the continuous-time signal can be expressed in terms of samples using simple Whittaker-Shannon interpolation, there is no closed-form expression relating the samples generated by MWC and the continuous-time signal. In order to reconstruct the signal, we require compressive sensing (CS) algorithm. The CS algorithm, simultaneous orthogonal matching pursuit (SOMP) used in the reconstruction stage requires a minimum rate of 4N B log(M/2N), which is nearly twice the theoretical rate. In this paper, we propose a new greedy algorithm, which exploits the clustered sparse structure of the multiband signals to sample at near-theoretical rates. The simulation results supporting the better performance of our algorithm are also presented.