  In this work, we study the phase-space and chemical properties of Sagittarius
(Sgr) stream, the tidal tails produced by the ongoing destruction of Sgr dwarf
spheroidal (dSph) galaxy, focusing on its very metal-poor (VMP; $\rm[Fe/H] <
-2$) content. We combine spectroscopic and astrometric information from SEGUE
and $Gaia$ EDR3, respectively, with data products from a new large-scale run of
$\texttt{StarHorse}$ spectro-photometric code. Our selection criteria yields
${\sim}1600$ stream members, including ${>}200$ VMP stars. We find the leading
arm ($b>0^\circ$) of Sgr stream to be more metal-poor, by ${\sim}0.2$ dex, than
the trailing one ($b<0^\circ$). With a subsample of turnoff and subgiant stars,
we estimate this substructure's stellar population to be ${\sim}1$ Gyr older
than the thick disk's. With the aid of an $N$-body model of the Sgr system, we
verify that simulated particles stripped earlier (${>}2$ Gyr ago) have
present-day phase-space properties similar to lower-metallicity stream stars.
Conversely, those stripped more recently (${<}2$ Gyr) are preferentially more
akin to metal-rich ($\rm[Fe/H] > -1$) members of the stream. Such correlation
between kinematics and chemistry can be explained by the existence of a
dynamically hotter, less centrally-concentrated, and more metal-poor population
in Sgr dSph prior to its disruption, implying that this galaxy was able to
develop a metallicity gradient before its accretion. Finally, we discovered
several carbon-enhanced metal-poor ($\rm[C/Fe] > +0.7$ and $\rm[Fe/H] \leq
-1.5$) stars in Sgr stream, which is in tension with current observations of
its remaining core where such objects are not found.
