Wideband spectrum sensing based on sub-Nyquist sampling is an attractive approach to propel Cognitive Radios (CRs), which aims to alleviate the burden of sampling and improve the utilization of frequency resources. In the context of compressive sensing (CS), finding idle frequency bands can be equivalent to calculating the complement of support set in a constructed multiple measurement vectors (MMV) problem. To ensure the performance of support set recovery in noisy environments, most existing algorithms require the prior knowledge of signal sparsity order, which is an unknown time-varying variable. To address the dependence of recovery performance on signal sparsity order, a two-step solution for wideband spectrum sensing in the Modulated Wideband Converter (MWC) sub-Nyquist sampling architecture is proposed in this paper. The proposed solution will first estimate the signal sparsity order from the compressed covariance matrix through state-of-the-art Model Order Selection (MOS) techniques, and then use the estimated sparsity order to adjust the support set recovery algorithms. Moreover, the relationship between the rank of the compressed covariance matrix and the signal sparsity order is provided through theoretical and numerical analysis. Simulations verify that the proposed solution can effectively reduce the probability of false alarms to meet the stringent sensing requirements of IEEE 802.22 in the medium sparse CR network.