We present a strong lensing analysis on the massive cluster Abell 370 (A370; z=0.375), using a combination of deep multiband Hubble Space Telescope (HST) imaging and Multi-Unit Spectroscopic Explorer (MUSE) spectroscopy. From only 2 h of the MUSE data, we are able to measure 120 redshifts in the southern BCG area, including several multiply imaged lens systems. In total, we increase the number of multiply imaged systems with a secure redshift from 4 to 15, nine of which are newly discovered. Of these, eight are located at z>3, greatly extending the redshift range of spectroscopically confirmed systems over previous work. Using these systems as constraints, we update a parametric lens model of A370, probing the mass distribution from cluster to galaxy scales. Overall, we find that a model with only two cluster-scale dark matter haloes (one for each BCG) does a poor job of fitting these new image constraints. Instead, two additional mass clumps - a central 'bar' of mass located between the BCGs, and another clump located within a 'crown' of galaxies in the northern part of the cluster field - provide significant improvements to the fit. Additional physical evidence suggests these clumps are indeed real features of the system, but with relatively few image constraints in the crown region, this claim is difficult to evaluate from a modelling perspective. Additional MUSE observations of A370 covering the entire strong-lensing region will greatly help these efforts, further improving our understanding of this intriguing cluster.