  In this work, the $f(\mathcal{G},T)$ theory of gravity is recast in terms of
the $\phi$ and $\psi$ fields within the scalar-tensor formulation, where
$\mathcal{G}$ is the Gauss-Bonnet term and $T$ denotes the trace of the
energy-momentum tensor. The general aspects of the introduced reformulation are
discussed and the reconstruction of the cosmological scenarios is presented,
focusing on the so-called hybrid evolution. As a result, the scalar-tensor
$f(\mathcal{G},T)$ theory is successfully reconstructed for the early and late
time approximations with the corresponding potentials. The procedure of
recovering the $f(\mathcal{G},T)$ theory in the original formulation is
performed for the late time evolution and a specific quadratic potential. The
scalar-tensor formulation introduced herein not only facilitates the
description of various cosmic phases but also serves as a viable alternative
portrayal of the $f(\mathcal{G},T)$ gravity which can be viewed as an extension
of the well-established scalar Einstein-Gauss-Bonnet gravity.
