This paper addresses the reconstruction of sea surface currents from satellite ocean sensing data. Whereas the classical surface currents derived from the SSH (Sea Surface Height) products are rather low space-time resolution fields (typically, 50 km and 12-day actual space-time grid resolution), we investigate the extent to which we can retrieve sea surface currents at higher resolution using daily SST (Sea Surface Temperature) satellite observations. State-of-the-art methods, which exploit classical optical flow schemes or nonlinear regression techniques, do not provide satisfactory results due to the space-time variabilities of the relationships between the SST and the sea surface current. Motivated by our recent joint SST-SSH identification of characterization of upper ocean dynamical modes, we here show that a multi-regime model, formally stated as a Markov-switching latent class regression model, provides a relevant model to capture the above-mentioned variabilities and reconstruct SST-driven sea surface currents. The considered case study within the Agulhas current demonstrates that our model retrieves high-resolution space-time details which cannot be resolved by the classical SSH-derived products.