  We use zoom-in, hydrodynamical, cosmological $N$-body simulations tracing the
formation of the first stellar clumps from the SImulating the Environments
where Globular clusters Emerged (SIEGE) project, to study key structural
properties of dark matter haloes when the Universe was only $0.92$ Gyr old. The
very high-resolution (maximum physical resolution 0.3 h$^{-1}$ pc at $z=6.14$,
smallest dark-matter particle mass $164\,M_{\odot}$) allows us to reach the
very low mass end of the stellar-to-halo mass relation ($M_{\rm
vir}=10^{7.5-9.5}\,M_{\odot}$) to study the processes that mould dark matter
haloes during the first stages of structure formation. We investigate the role
of baryonic cooling and stellar feedback, modeled from individual stars, in
shaping haloes, and of environmental effects as accretion of dark matter along
cosmic filaments and mergers. We find that the onset of star formation
(typically for $\log M_{\rm vir}/M_{\odot}\simeq7.6$) causes the inner cusp in
the haloes density profile to flatten into a core with constant density and
size proportionally to the halo virial mass. Even at these mass scales, we
confirm that baryons make haloes that have formed stars rounder in the central
regions than haloes that have not formed stars yet, with median minor-to-major
$\langle q \rangle$ and intermediate-to-major $\langle s \rangle$ axes 0.66 and
0.84, respectively. Our morphological analysis shows that, at $z=6.14$, haloes
are largely prolate in the outer parts, with the major axis aligned along
filaments of the cosmic web or towards smaller sub-haloes, with the degree of
elongation having no significant dependence on the halo mass.
