  We examine extended inflation models enhanced by the addition of a coupling
between the inflaton field and the space-time curvature. We examine two types
of model, where the underlying inflaton potential takes on second-order and
first-order form respectively. One aim is to provide models which satisfy the
solar system constraints on the Brans--Dicke parameter $\omega$. This
constraint has proven very problematic in previous extended inflation models,
and we find circumstances where it can be successfully evaded, though the
constraint must be carefully assessed in our model and can be much stronger
than the usual $\omega > 500$. In the simplest versions of the model, one may
avoid the need to introduce a mass for the Brans--Dicke field in order to
ensure that it takes on the correct value at the present epoch, as seems to be
required in hyperextended inflation. We also briefly discuss aspects of the
formation of topological defects in the inflaton field itself.
