  This paper presents the third data release of the INvestigating Stellar
Population In RElics (INSPIRE) project, comprising 52 ultra-compact massive
galaxies (UCMGs) observed with the X-Shooter spectrograph. We measure
integrated stellar velocity dispersion, [Mg/Fe] abundances, ages, and
metallicities for all the INSPIRE objects. We thus infer star formation
histories and confirm the existence of a degree of relicness (DoR), defined in
terms of the fraction of stellar mass formed by $z=2$, the time at which a
galaxy has assembled 75\% of its mass, and the final assembly time. Objects
with a high DoR assembled their stellar mass at early epochs, while low-DoR
objects show a non-negligible fraction of later-formed populations and hence a
spread in ages and metallicities. A higher DoR correlates with larger [Mg/Fe],
super-solar metallicity, and larger velocity dispersion values. The 52 UMCGs
span a large range of DoR from 0.83 to 0.06, with 38 of them having formed more
than 75\% of their mass by $z=2$. Of these, nine are extreme relics
(DoR$>0.7$), since they formed the totality ($>99\%$) of their stellar mass by
redshift $z=2$. The remaining 14 UCMGs cannot be considered relics, as they are
characterised by more extended star formation histories. With INSPIRE, we built
the first sizeable sample of relics outside the local Universe, up to
$z\sim0.4$, increasing the number of confirmed relics by a factor of $>10$, and
opening up an important window to explain the mass assembly of massive galaxies
in the high-z Universe.
