  We study the impact of thermalization and number-changing processes in the
dark sector on the yield of gravitationally produced dark matter (DM). We take
into account the DM production through the $s$-channel exchange of a massless
graviton both from the scattering of inflatons during the reheating era, and
from the Standard Model bath via the UV freeze-in mechanism. By considering the
DM to be a scalar, a fermion, and a vector boson we show, in a
model-independent way, that DM self-interaction gives rise to a larger viable
parameter space by allowing lower reheating temperature to be compatible with
Planck observed relic abundance. As an example, we also discuss our findings in
the ontext of the $\mathbb{Z}_2$-symmetric scalar singlet DM model.
