The spin-structure functions $g_1$ and $g_2$, and the spin-dependent partial cross-section $��_\mathrm{TT}$ have been extracted from the polarized cross-sections differences, $����_{\parallel}\hspace{-0.06cm}\left(��,Q^{2}\right)$ and $����_{\perp}\hspace{-0.06cm}\left(��,Q^{2}\right)$ measured for the $\vec{^\textrm{3}\textrm{He}}(\vec{\textrm{e}},\textrm{e}')\textrm{X}$ reaction, in the E97-110 experiment at Jefferson Lab. Polarized electrons with energies from 1.147 to 4.404 GeV were scattered at angles of 6$^{\circ}$ and 9$^{\circ}$ from a longitudinally or transversely polarized $^{3}$He target. The data cover the kinematic regions of the quasi-elastic, resonance production and beyond. From the extracted spin-structure functions, the first moments $\overline{��_1}\hspace{-0.06cm}\left(Q^{2}\right)$, $��_2\hspace{-0.06cm}\left(Q^{2}\right)$ and $I_{\mathrm{TT}}\hspace{-0.06cm}\left(Q^{2}\right)$ are evaluated with high precision for the neutron in the $Q^2$ range from 0.035 to 0.24~GeV$^{2}$. The comparison of the data and the chiral effective field theory predictions reveals the importance of proper treatment of the $��$ degree of freedom for spin observables. 7 pages, 5 figures, 3 tables. Version published in Phys. Lett. B
