  We examine the effects of a stiff pre-recombination era on the present day's
energy spectrum of the primordial gravitational waves. If the background total
equation of state parameter at the pre-recombination era is described by a
kination era one, this directly affects the modes with characteristic
wavenumbers which reenter the Hubble horizon during this stiff era. The stiff
era causes a broken-power-law effect on the energy spectrum of the
gravitational waves. We use two approaches, one model agnostic and a specific
model that can realize this scenario. In all cases, the inflationary era can be
realized either by some theory leading to a standard red-tilted tensor spectral
index or by some theory which has a mild tensor spectral index
$n_{\mathcal{T}}=0.17-0.37$ like an Einstein-Gauss-Bonnet theory. For the model
agnostic scenario case, the NANOGrav signal can be explained by the stiff
pre-recombination era combined with an inflationary era with a mild blue-tilted
tensor spectral index $n_{\mathcal{T}}=0.37$ and a low-reheating temperature
$T_R\sim 0.1$GeV. In the same case, the red-tilted inflationary theory signal
can be detectable by the future LISA, BBO and DECIGO experiments. The model
dependent approach is based on a Higgs-axion model which can yield multiple
deformations of the background total equation of state parameter, causing
multiple broken-power-law behaviors occurring in various eras before and after
the recombination era. In this case, the NANOGrav signal is explained by this
model in conjunction with an inflationary era with a really mild blue-tilted
tensor spectral index $n_{\mathcal{T}}=0.17$ and a low-reheating temperature
$T_R\sim 20\,$GeV. In this case, the signal can be detectable by the future
Litebird experiment, which is a very characteristic pattern in the tail of the
primordial gravitational wave energy spectrum.
