Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid and assume for simplicity that $T$ is countable.

If we want to restrict our attention to models $\mathcal{M}$ of $T$ for which $\leq$ actually well-orders $M$ we immediately get a kind of downward Löwenheim–Skolem theorem since any sub-order of a well-order is automatically well-ordered. The naive analog of the upward Löwenheim–Skolem theorem is false because the only well-ordered model of $\text{Th}(\mathbb{N})$ is the standard one, but there is still the possibility of analogous statements requiring the existence of uncountable well-ordered models.

Ultraproducts of well-ordered sets with regards to countably complete ultrafilters are well-ordered, so assuming that they exist (i.e. assuming the existence of a measurable cardinal) and assuming $T$ has a well-ordered model whose cardinality can be increased by a countably complete ultrapower, then we get a strictly larger well-ordered model. (I assume that this works for all uncountable cardinals or maybe for all cardinals with uncountable cofinality?) It doesn't necessarily follow that we have arbitrarily large well-ordered models because unions of elementary chains of well-ordered sets aren't necessarily well-ordered, unless countably complete ultrapowers of ordinals are always end-extensions, which I don't know one way or the other.

There are a lot of questions: Is there a sufficient condition for the existence of arbitrarily large well-ordered models of $T$ (assuming the existence of a measurable cardinal if necessary)? Is there a way of doing it without measurable cardinals? Is it possible to characterize which order types well-ordered models of $T$ can be? When do models of $T$ have proper elementary end extensions? (And incidentally when do well-ordered models of $T$ have initial segments that are elementary sub-structures?)