  Natural Inflation with non-minimal coupling (NMC) to gravity, embodied by a
Lagrangian term $\xi \phi^2 R $, is investigated in the context of an extended
gravity of the form $R+ \alpha R^2$. The treatment is performed in the Palatini
formalism. We discuss various limits of the model ``$\alpha \gg 1$'' and
``$\alpha \ll 1$'' in light of two scenarios of inflation: a ``Slow roll'' and
a ``Constant roll'' scenario. By analyzing the observational consequences of
the model, our results show a significant improvement regarding compatibility
between the theoretical results of this model and the observational constraints
from Planck 2018 and BICEP/Keck 2018, as exemplified by the tensor-to-scalar
ratio and spectral index. Furthermore, a broader range for the parameter space
of natural inflation is now compatible with the confidence contours of Planck
\& BICEP/Keck results. The joint effects of the contributions of both the NMC
to gravity and the $\alpha R^2$ make a significant improvement: $\alpha R^2$
gravity influences scalar-tensor ratio values, whereas NMC to gravity has a
more significant impact on the spectral index values. Contributions from both
terms allow more previously excluded intervals to be included being compatible
now with observational data. These conclusions about the roles of NMC to
gravity and, particularly, the extended gravity remain mainly valid with a
periodic NMC similar in form to the natural inflation potential.
