In this paper, we address the problem of detecting and localizing closely spaced moving targets using a compressed sensing approach. In a previous work, we demonstrated the interest of a compressed sensing approach for overcoming Doppler ambiguities and proposed the N-signal Adaptive Grid Orthogonal Matching Pursuit (N-signal-AGOMP) [1] for the detection and estimation of moving targets in a continuous and ambiguous Doppler domain. Such an approach has to face grid refinement problems that may lead to reduced estimation performance when targets are closely spaced. In this paper, we propose a high resolution Orthogonal Matching Pursuit (OMP) based algorithm that sequentially refines target parameters based on a Non-linear Least Squares (NLS) estimator. We also propose to combine this algorithm with a hypothesis test process to enhance the probability to distinguish nearby targets. Simulation results show that the proposed method allows to detect and accurately estimate target parameters even when they are separated by less than half the Doppler Matched Filter (MF) resolution.