We use the Beyond Ultra-deep Frontier Fields and Legacy Observations (BUFFALO) strong lensing image catalogue of the merging galaxy cluster Abell 370 to obtain a mass model using free-form lens inversion algorithm grale. The improvement of the strong lensing data quality results in a lens plane rms of only 0.45 arcsec, about a factor of two lower than that of our existing HFF v4 reconstruction. We attribute the improvement to spectroscopic data and use of the full reprocessed HST mosaics. In our reconstructed mass model, we found indications of three distinct mass features in Abell 370: (i) a similar to 35 kpc offset between the northern BCG and the nearest mass peak, (ii) a similar to 100 kpc mass concentration of roughly critical density similar to 250 kpc east of the main cluster, and (iii) a probable filament-like structure passing N-S through the cluster. While (i) is present in some form in most publicly available reconstructions spanning the range of modelling techniques: parametric, hybrid, and free-form, (ii) and (iii) are recovered by only about half of the reconstructions. We tested our hypothesis on the presence of the filament-like structure by creating a synthetic cluster - Irtysh IIIc - mocking the situation of a cluster with external mass. We also computed the source plane magnification distributions. Using them, we estimated the probabilities of magnifications in the source plane, and scrutinized their redshift dependence. Finally, we explored the lensing effects of Abell 370 on the luminosity functions of sources at z(s) = 9.0, finding it consistent with published results.