Context. The compact radio source Sagittarius A^∗ (Sgr A^∗) in the Galactic centre is the primary supermassive black hole candidate. General relativistic magnetohydrodynamical (GRMHD) simulations of the accretion flow around Sgr A^∗ predict the presence of sub-structure at observing wavelengths of ∼3 mm and below (frequencies of 86 GHz and above). For very long baseline interferometry (VLBI) observations of Sgr A^∗ at this frequency the blurring effect of interstellar scattering becomes sub-dominant, and arrays such as the high sensitivity array (HSA) and the global mm-VLBI array (GMVA) are now capable of resolving potential sub-structure in the source. Such investigations help to improve our understanding of the emission geometry of the mm-wave emission of Sgr A^∗, which is crucial for constraining theoretical models and for providing a background to interpret 1 mm VLBI data from the Event Horizon Telescope (EHT). Aims. Following the closure phase analysis in our first paper, which indicates asymmetry in the 3 mm emission of Sgr A^∗, here we have used the full visibility information to check for possible sub-structure. We extracted source size information from closure amplitude analysis, and investigate how this constrains a combined fit of the size-frequency relation and the scattering law for Sgr A^∗. Methods. We performed high-sensitivity VLBI observations of Sgr A^∗ at 3 mm using the Very Long Baseline Array (VLBA) and the Large Millimeter Telescope (LMT) in Mexico on two consecutive days in May 2015, with the second epoch including the Greenbank Telescope (GBT). Results. We confirm the asymmetry for the experiment including GBT. Modelling the emission with an elliptical Gaussian results in significant residual flux of ∼10 mJy in south-eastern direction. The analysis of closure amplitudes allows us to precisely constrain the major and minor axis size of the main emission component. We discuss systematic effects which need to be taken into account. We consider our results in the context of the existing body of size measurements over a range of observing frequencies and investigate how well-constrained the size-frequency relation is by performing a simultaneous fit to the scattering law and the size-frequency relation. Conclusions. We find an overall source geometry that matches previous findings very closely, showing a deviation in fitted model parameters less than 3% over a time scale of weeks and suggesting a highly stable global source geometry over time. The reported sub-structure in the 3 mm emission of Sgr A^∗ is consistent with theoretical expectations of refractive noise on long baselines. However, comparing our findings with recent results from 1 mm and 7 mm VLBI observations, which also show evidence for east-west asymmetry, we cannot exclude an intrinsic origin. Confirmation of persistent intrinsic substructure will require further VLBI observations spread out over multiple epochs.