Compressed sensing (CS)-based wideband spectrum sensing approaches have attracted much attention because they release the burden of high signal acquisition costs. However, in CS-based sensing approaches, highly non-linear reconstruction methods are used for spectrum recovery, which require high computational complexity. This letter proposes a two-step compressive wideband sensing algorithm. This algorithm introduces a coarse sensing step to further compress the sub-Nyquist measurements before spectrum recovery in the following compressive fine sensing step, as a result of the significant reduction in computational complexity. Its enabled sufficient condition and computational complexity are analyzed. Even when the sufficient condition is just satisfied, the average reduced ratio of computational complexity can reach 50% compared with directly performing compressive sensing with the excellent algorithm that is used in our fine sensing step.