  Natural inflation is an attractive model for primordial inflation, since the
potential for the inflaton is of the pseudo Nambu-Goldstone form,
$V(\phi)=\Lambda^4 [1+\cos (\phi/f)]$, and so is protected against radiative
corrections. Successful inflation can be achieved if $f \gtrsim {\rm few}\,
M_{P}$ and $\Lambda \sim m_{GUT}$ where $\Lambda$ can be seen as the strong
coupling scale of a given non-abelian gauge group. However, the latest
observational constraints put natural inflation in some tension with data. We
show here that a non-minimal coupling to gravity $\gamma^2(\phi) R$, that
respects the symmetry $\phi\rightarrow \phi+2 \pi f$ and has a simple form,
proportional to the potential, can improve the agreement with cosmological
data. Moreover, in certain cases, satisfactory agreement with the Planck 2018
TT, TE, EE and low P data can be achieved even for a periodicity scale of
approximately $M_p$.
