Satellite-based remote sensing missions have revolutionized our understanding of the Ocean state and dynamics. Among them, spaceborne altimetry provides valuable measurements of Sea Surface Height (SSH), which is used to estimate surface geostrophic currents. However, due to the sensor technology employed, important gaps occur in SSH observations. Complete SSH maps are produced by the altimetry community using linear Optimal Interpolations (OI) such as the widely-used Data Unification and Altimeter Combination System (DUACS). However, OI is known for producing overly smooth fields and thus misses some mesostructures and eddies. On the other hand, Sea Surface Temperature (SST) products have much higher data coverage and SST is physically linked to geostrophic currents through advection. We design a realistic twin experiment to emulate the satellite observations of SSH and SST to evaluate interpolation methods. We introduce a deep learning network able to use SST information, and a trainable in two settings: one where we have no access to ground truth during training and one where it is accessible. Our investigation involves a comparative analysis of the aforementioned network when trained using either supervised or unsupervised loss functions. We assess the quality of SSH reconstructions and further evaluate the network's performance in terms of eddy detection and physical properties. We find that it is possible, even in an unsupervised setting to use SST to improve reconstruction performance compared to SST-agnostic interpolations. We compare our reconstructions to DUACS's and report a decrease of 41\% in terms of root mean squared error.