  We give a formalism for constructing hidden sector bundles as extensions of
sums of line bundles in heterotic $M$-theory. Although this construction is
generic, we present it within the context of the specific Schoen threefold that
leads to the physically realistic $B-L$ MSSM model. We discuss the embedding of
the line bundles, the existence of the extension bundle, and a number of
necessary conditions for the resulting bundle to be slope-stable and thus $N=1$
supersymmetric. An explicit example is presented, where two line bundles are
embedded into the $SU(3)$ factor of the $E_{6} \times SU(3)$ maximal subgroup
of the hidden sector $E_{8}$ gauge group, and then enhanced to a non-Abelian
$SU(3)$ bundle by extension. For this example, there are in fact six
inequivalent extension branches, significantly generalizing that space of
solutions compared with hidden sectors constructed from a single line bundle.
