We use H i absorption along the lines-of-sight to the Q0107 quasar triplet in order to model potential disc and outflow structures in the circumgalactic medium of intervening galaxies at z less than or similar to 1, as well as the intergalactic medium on scales of up to a few virial radii. We consider a sample of twelve isolated galaxies in the Q0107 field with position angles and inclinations measured from HST imaging as well as redshifts from our spectroscopic surveys, alongside 27 detected Ly alpha absorbers within 500 km s(-1) of these galaxies. Building on previous work showing increased incidence of absorption close to the projected major and minor axes, we use model rotating discs and bi-conical outflows in attempting to reproduce the observed absorption. Requiring these models to match absorption in multiple lines-of-sight provides additional constraints over single-sightline observations. We identify four possible outflows with velocities similar to 100 km s(-1), two of which extend to or beyond the virial radius, with a variety of opening angles. Two galaxies have nearby co-rotating absorbers with rotation velocities less than or similar to v(vir), that may probe disc-like structures, and we can rule out a disc/outflow origin for a further ten absorbers. These indicate that outflowing and co-rotating structures can extend to large scales but are either not ubiquitous, or do not always produce detectable Ly alpha. In some cases, disc models are successful even close to the minor axis of the galaxy, and some of our model outflows exhibit wide opening angles. These results imply that purely geometrical cuts are not sufficient to distinguish between discs and outflows in single line-of-sight studies.