The antiferromagnetic insulator α-Fe2O3 (hematite), widely used in spintronics and magnonics, features a spin-reorientation transition (Morin transition) at 263 K. Thin films, however, often lack this Morin transition, limiting their potential applications. Here, we investigate the impact of different growth conditions on the magnetic anisotropy in α-Fe2O3 films to tune the Morin transition temperature. To this end, we compare the structural, magnetic, and magnon-based spin transport properties of α-Fe2O3 films with different thicknesses grown by pulsed laser deposition in molecular and atomic oxygen atmospheres. We observe a finite Morin transition for those grown by atomic-oxygen-assisted deposition, interestingly even down to 19 nm thickness, where we find a Morin transition at 125 K. In easy-plane antiferromagnets, the nature and time-evolution of the elementary excitations of the spin system are captured by the orientation and precession of the magnon pseudospin around its equilibrium pseudofield, manifesting itself in the magnon Hanle effect. We characterize this effect in these α-Fe2O3 films via all-electrical magnon transport measurements. The films grown with atomic oxygen show a markedly different magnon spin signal from those grown in molecular oxygen atmospheres. Most importantly, the maximum magnon Hanle signal is significantly enhanced, and the Hanle peak is shifted to lower magnetic field values for films grown with atomic oxygen, suggesting changes in the magnetic anisotropy due to an increased oxygen content in these films. Our findings provide new insights into the possibility to fine-tune the magnetic anisotropy in α-Fe2O3 and thereby to engineer the magnon Hanle effect.