We present an updated strong-lensing analysis of the massive cluster Abell 370 (A370), continuing the work first presented in L17. In this new analysis, we take advantage of the deeper imaging data from the Hubble Space Telescope Frontier Fields programme, as well as a large spectroscopic mosaic obtained with the Multi-Unit Spectroscopic Explorer (MUSE). Thanks to the extended coverage of this mosaic, we probe the full 3D distribution of galaxies in the field, giving us a unique picture of the extended structure of the cluster and its surroundings. Our final catalogue contains 584 redshifts, representing the largest spectroscopic catalogue of A370 to date. Constructing the model, we measure a total mass distribution that is quantitatively similar to our previous work - though to ensure a low rms error in the model fit, we invoke a significantly large external shear term. Using the redshift catalogue, we search for other bound groups of galaxies, which may give rise to a more physical interpretation of this shear. We identify three structures in narrow redshift ranges along the line of sight, highlighting possible infalling substructures into the main cluster halo. We also discover additional substructure candidates in low-resolution imaging at larger projected radii. More spectroscopic coverage of these regions (pushing close to the A370 virial radius) and more extended, high-resolution imaging will be required to investigate this possibility, further advancing the analysis of these interesting developments.