  The excitation of trapped inertial waves (r-modes) by warps and
eccentricities in the inner regions of a black hole accretion disc may explain
the high-frequency quasi-periodic oscillations (HFQPOs) observed in the
emission of Galactic X-ray binaries. However, it has been suggested that strong
vertical magnetic fields push the oscillations' trapping region toward the
innermost stable circular orbit (ISCO), where conditions could be unfavourable
for their excitation. This paper explores the effects of large-scale magnetic
fields that exhibit \textit{both} toroidal and vertical components, through
local and global linear analyses. We find that a strong toroidal magnetic field
can reduce the detrimental effects of a vertical field: in fact, the isolation
of the trapping region from the ISCO may be restored by toroidal magnetic
fields approaching thermal strengths. The toroidal field couples the r-modes to
the disc's magneto-acoustic response and inflates the effective pressure within
the oscillations. As a consequence, the restoring force associated with the
vertical magnetic field's tension is reduced. Given the analytical and
numerical evidence that accretion discs threaded by poloidal magnetic field
lines develop a strong toroidal component, our result provides further evidence
that the detrimental effects of magnetic fields on trapped inertial modes are
not as great as previously thought.
