  We consider 5-manifolds with a contact form arising from a hypo structure,
which we call \emph{hypo-contact}. We provide conditions which imply that there
exists such a structure on an oriented hypersurface of a 6-manifold with a
half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field
$X$ preserving the SU(3)-structure we study the geometry of the orbits space.
Moreover, we describe the solvable Lie algebras admitting a \emph{hypo-contact}
structure. This allows us exhibit examples of Sasakian $\eta$-Einstein
manifolds, as well as to prove that such structures give rise to new metrics
with holonomy SU(3) and to new metrics with holonomy $G_2$.
