  Understanding the structure of indecomposable $n$-dimensional persistence
modules is a difficult problem, yet is foundational for studying
multipersistence. To this end, Buchet and Escolar showed that any finitely
presented rectangular $(n-1)$-dimensional persistence module with finite
support is a hyperplane restriction of an $n$-dimensional persistence module.
We extend this result to the following: If $M$ is any finitely presented
$(n-1)$-dimensional persistence module with finite support, then there exists
an indecomposable $n$-dimensional persistence module $M'$ such that $M$ is the
restriction of $M'$ to a hyperplane. We also show that any finite zigzag
persistence module is the restriction of some indecomposable $3$-dimensional
persistence module to a path.
