  In easy-plane antiferromagnets, the nature of the elementary excitations of
the spin system is captured by the precession of the magnon pseudospin around
its equilibrium pseudofield, manifesting itself in the magnon Hanle effect.
Here, we investigate the impact of growth-induced changes in the magnetic
anisotropy on this effect in the antiferromagnetic insulator
$\alpha$-Fe$_2$O$_3$ (hematite). To this end, we compare the structural,
magnetic, and magnon-based spin transport properties of $\alpha$-Fe$_2$O$_3$
films with different thicknesses grown by pulsed laser deposition in molecular
and atomic oxygen atmospheres. While in films grown with molecular oxygen a
spin-reorientation transition (Morin transition) is absent down to $10\,$K, we
observe a Morin transition for those grown by atomic-oxygen-assisted
deposition, indicating a change in magnetic anisotropy. Interestingly, even for
a $19\,$nm thin $\alpha$-Fe$_2$O$_3$ film grown with atomic oxygen we still
detect a Morin transition at $125\,$K. We characterize the magnon Hanle effect
in these $\alpha$-Fe$_2$O$_3$ 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. These observations suggest a change of magnetic anisotropy for
$\alpha$-Fe$_2$O$_3$ films fabricated by atomic-oxygen-assisted deposition
resulting in an increased oxygen content in these films. Our findings provide
new insights into the possibility to fine-tune the magnetic anisotropy in
$\alpha$-Fe$_2$O$_3$ and thereby to engineer the magnon Hanle effect.
