  We develop the calculation of the divergent part of one-loop covariant
effective action for scalar fields minimally and non-minimally coupled to
gravity using the generalized Schwinger-DeWitt technique. We derive the
field-space metric using Vilkovisky's prescription and obtain new terms in the
one-loop corrections which are absent in past literature with trivial choices
of field-space metric. We further calculate the covariant versions of past
results, obtained using the standard approach, by applying Barvinsky and
Vilkovisky's extension to the ordinary Schwinger-DeWitt approach. For
completeness, we study the one-loop gravitational corrections for a
dilaton-extended two-field Starobinsky model and compare with known results.
