  We provide a detailed study of natural inflation with a periodic non-minimal
coupling, which is a well-motivated inflationary model that admits an explicit
UV completion. We demonstrate that this construction can satisfy the most
recent observational constraints from Planck and the BICEP/Keck collaborations.
We also compute the corresponding relic gravitational wave background due to
tensor perturbations and show that future space-borne interferometers, such as
DECIGO, BBO and ALIA, may be able to detect it. Next, we extend this analysis
and establish the validity of these results in a multi-field model featuring an
additional $R^2$ term in the action, which allows us to interpolate between
natural and scalaron (a.k.a.~Starobinsky) inflation. We investigate the
conditions under which the aforementioned future interferometers will have the
capability to differentiate between pure natural inflation and natural-scalaron
inflation. The latter analysis could open the door to distinguishing between
single-field and multi-field inflation through gravitational wave observations
in more general contexts.
