Let $\mathbb{T}$ be a multi-sorted algebraic theory (i.e. a set of equations between terms over a multi-sorted signature with function symbols and constants). In particular, I am interested in the case where $\mathbb{T}$ has at least two sorts. For any model $M$ of $\mathbb{T}$ and any sort $A$ of $\mathbb{T}$, let $M_A$ be the set that interprets this sort in $M$.
I am interested in the following question: is there a condition that can be put on $\mathbb{T}$ to guarantee that if $M$ is any model of $\mathbb{T}$ such that $|M_A| \leq 1$ for some sort $A$, then there will be a model $N$ of $\mathbb{T}$ such that $|N_A| \geq 2$ and there is a homomorphism $h : M \to N$?
Or, to state the question in another (not necessarily equivalent) way, if $M\langle x_A \rangle$ is the coproduct of $M$ with the free $\mathbb{T}$-model on one generator $x_A$ of sort $A$, can we impose some condition on $\mathbb{T}$ to guarantee that $|M\langle x_A \rangle_A| \geq 2$?
For example, would it be sufficient to assume that $\mathbb{T}$ does not prove that the sort $A$ is trivial, in the sense that $\mathbb{T} \not\vdash x = x'$ for distinct variables $x, x'$ of sort $A$?