Say that a class $\mathcal{C}$ of countable first-order structures in some finite signature has the *effective substructure property* if $\mathcal{C}$ is closed under isomorphism and whenever $A\in \mathcal{C}$ has domain $\subseteq\omega$ and $B\in\mathcal{C}$ is isomorphic to a substructure of $A$, then $B$ has a presentation computable relative to $A$. My question is: what are some natural classes of structures with the effective substructure property, and what is a good source to learn about this property?

Some examples. The ur-example is the class $\mathcal{W}$ of all countable well-orders. $\mathcal{W}$ has the effective substructure property since $B\subsetneq A\in \mathcal{W}$ implies that $B$ is isomorphic to $\lbrace a\in A: a < b\rbrace$ for some $b\in A$, and this latter set is computable relative to $A$. Conversely, the collection of dense linear orders has the effective substructure property since all such linear orders are computable. Basically, where I'm at right now is that the only non-trivial example of a class with the effective substructure property I can think of is $\mathcal{W}$ (or variants). I'm wondering, in particular, if there is some reasonably natural class of structures with the effective substructure property which doesn't just build off of $\mathcal{W}$? For example, some $\mathcal{C}$ such that $\mathcal{W}$ is not Borel reducible to $\mathcal{C}$?