Natural inflation is an attractive model for primordial inflation, since the potential for the inflaton is of the pseudo Nambu-Goldstone form, V(ϕ)=Λ^4 [1+cos (ϕ/f)], and so is protected against radiative corrections. Successful inflation can be achieved if f ≳ few M_P and Λ∼ m_GUT where Λ 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

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... ing to gravity γ^2(ϕ) R, that respects the symmetry ϕ→ϕ+2 π 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.