In Hirschhorn's "Model Categories and Their Localizations", section 15.7, there's the following corollary to the preceding Proposition:

$\mathbb{Corollary}$ 15.7.2 If $\mathfrak{M}$ is a cellular model category and $\mathfrak{C}$ is a Reedy category, then the cofibrations of the Reedy model category on $\mathfrak{M}^{\mathfrak{C}}$ are monomorphisms.

I think we can use this to prove the following: If we've got a Reedy model category $\mathfrak{A}$ and a category $\mathcal{B}$ in which all cofibrations are monomorphisms and weak equivalences are pointwise (i.e. just a rough way to say that it's equipped with an injective model structure) then in the category $\text{Psh}(\mathfrak{A},\text{Psh}(\mathcal{B},\textit{Ssets}))$ all Reedy cofibrations are monomorphisms.

Is there a way to have the converse work for some types of Reedy model categories? (By converse I mean all monomorphisms being Reedy cofibrations)

If I'm not mistaken, this will imply some sort of relation between model structures.

Edit: Big thanks to Simon for pointing out the vague, unexplicit nature of the question.

Cofibrations of Reedy model categorieswould be much more informative. $\endgroup$