In this paper, a reflective optical pump-probe technique for laser-driven plasmas at solid density target surfaces is presented. The technique is termed high depth-of-field time-resolved microscopy and it exploits the angular redistribution of the probe beam intensity after the probe's reflection from an expanded and hence non-planar iso-density surface in the plasma. The main application of the robust technique, which uses simple imaging of the probe beam, is the spatio-temporal resolution of the plasma formation and expansion at the target rear surface. Analytic and numerical modeling of the experimental setup are applied for the analysis of the experimental results. The relevance and potential of the optical plasma probing method is highlighted by the application to targets of different geometries, helping to understand the target shape-related differences in the ion acceleration performance.