The reconstruction of better-resolved sea surface currents is a key challenge in space oceanography. Besides the upcoming SWOT wide-swath altimeter mission, new algorithms are explore to produce improved gap-free gridded products. Based on the recent development of a generic end-to-end deep learning scheme for inverse problems backed on a variational formulation, we investigate how this framework applies to the space-time interpolation of satellite-derived SSH fields. We consider different parameterization of the proposed end-toend learning scheme, especially regarding the embedded variational solver. Using an Observing System Simulation Experiment based on high-resolution numerical simulations in the Gulf Stream region, we show that the later may significantly outperform the state-of-the-art, including DUACS optimal interpolation product, when jointly considering nadir along-track altimeter data and upcoming SWOT wide-swath data.