  We study T-duality chains of five-branes in heterotic supergravity where the
first order $\alpha'$-corrections are present. By performing the
$\alpha'$-corrected T-duality transformations of the heterotic NS5-brane
solutions, we obtain the KK5-brane and the exotic $5^2_2$-brane solutions
associated with the symmetric, the neutral and the gauge NS5-branes. We find
that the Yang-Mills gauge field in these solutions satisfies the self-duality
condition in the three- and two-dimensional transverse spaces to the brane
world-volumes. The $O(2,2)$ monodromy structures of the $5^2_2$-brane solutions
are investigated by the $\alpha'$-corrected generalized metric. Our analysis
shows that the symmetric $5^2_2$-brane solution, which satisfies the standard
embedding condition, is a T-fold and it exhibits the non-geometric nature. We
also find that the neutral $5^2_2$-brane solution is a T-fold at least at
$\mathcal{O} (\alpha')$. On the other hand, the gauge $5^2_2$-brane solution is
not a T-fold but show unusual structures of space-time.
