  Gravitational self-force theory is the primary way of modelling
extreme-mass-ratio inspirals (EMRIs). One difficulty that appears in
second-order self-force calculations is the strong divergence at the worldline
of the small object, which causes both numerical and analytical issues.
Previous work [Phys. Rev. D 95, 104056 (2017); ibid. 103, 124016 (2021)]
demonstrated that this could be alleviated within a class of highly regular
gauges and presented the metric perturbations in these gauges in a local
coordinate form. We build on this previous work by deriving expressions for the
highly regular gauge metric perturbations in both fully covariant form and as a
generic coordinate expansion. With the metric perturbations in covariant or
generic coordinate form, they can easily be expressed in any convenient
coordinate system. These results can then be used as input into a puncture
scheme in order to solve the field equations describing an EMRI.
