This paper is focused on the problem of the detection of multiple scatterers in synthetic aperture radar (SAR) tomography. The method presented exploits the a priori information that at most K_max different scatterers are present in the same range-azimuth resolution cell. In particular, a simplified version of a generalized-likelihood ratio test (GLRT) detector, based on support estimation (Sup-GLRT), is proposed. The Sup-GLRT is a constant false alarm rate sequential test that detects the presence of scatterers, one after another, and estimates their positions, detecting the support of the unknown signal. The proposed simplified test denoted as Fast-Sup-GLRT detector, despite still being a multistep statistical hypothesis test, exploits, at each step i, an approximated maximum-likelihood estimate of the signal support of cardinality i-1, based on the sequential estimation of i -1 supports of cardinality one. The introduced approximation allows a considerable reduction of the computational complexity, which from the combinatorial trend of Sup-GLRT passes to the linear one of Fast-Sup-GLRT, without significantly impairing the detection probability. The performance of the proposed approach is analyzed using TerraSAR-X system parameters, with particular reference to the elevation superresolution achievable for an assigned probability of false alarm and with a given number of acquisitions. Numerical results on simulated and real data are presented and discussed.