  In this paper, we explore higher-dimensional asymptotically flat wormhole
geometries in the framework of Gauss-Bonnet (GB) gravity and investigate the
effects of the GB term, by considering a specific radial-dependent redshift
function and by imposing a particular equation of state. This work is motivated
by previous assumptions that wormhole solutions were not possible for the $k=1$
and $\alpha < 0$ case, where $k$ is the sectional curvature of an
$(n-2)$-dimensional maximally symmetric space, and $\alpha$ is the Gauss-Bonnet
coupling constant. However, we emphasize that this discussion is purely based
on a nontrivial assumption that is only valid at the wormhole throat, and
cannot be extended to the entire radial-coordinate range. In this work, we
provide a counterexample to this claim, and find for the first time specific
solutions that satisfy the weak energy condition throughout the entire
spacetime, for $k=1$ and $\alpha < 0$. In addition to this, we also present
other wormhole solutions which alleviate the violation of the WEC in the
vicinity of the wormhole throat.
