  The expansion history of the Universe between the end of inflation and the
onset of radiation-domination (RD) is currently unknown. If the equation of
state during this period is stiffer than that of radiation, $w > 1/3$, the
gravitational wave (GW) background from inflation acquires a blue-tilt
${d\log\rho_{\rm GW}\over d\log f} = {2(w-1/3)\over (w+1/3)} > 0$ at
frequencies $f \gg f_{\rm RD}$ corresponding to modes re-entering the horizon
during the stiff-domination (SD), where $f_{\rm RD}$ is the frequency today of
the horizon scale at the SD-to-RD transition. We characterized in detail the
transfer function of the GW energy density spectrum, considering both 'instant'
and smooth modelings of the SD-to-RD transition. The shape of the spectrum is
controlled by $w$, $f_{\rm RD}$, and $H_{\rm inf}$ (the Hubble scale of
inflation). We determined the parameter space compatible with a detection of
this signal by LIGO and LISA, including possible changes in the number of
relativistic degrees of freedom, and the presence of a tensor tilt. Consistency
with upper bounds on stochastic GW backgrounds, however, rules out a
significant fraction of the observable parameter space. We find that this
renders the signal unobservable by Advanced LIGO, in all cases. The GW
background remains detectable by LISA, though only in a small island of
parameter space, corresponding to scenarios with an equation of state in the
range $0.46 \lesssim w \lesssim 0.56$ and a high inflationary scale $H_{\rm
inf} \gtrsim 10^{13}~{\rm GeV}$, but low reheating temperature $1~{\rm MeV}
\lesssim T_{\rm RD} \lesssim 150~{\rm MeV}$ (equivalently, $10^{-11}~{\rm Hz}
\lesssim f_{\rm RD} \lesssim 3.6\cdot10^{-9}~{\rm Hz}$). Implications for early
Universe scenarios resting upon an SD epoch are briefly discussed.
