$\Sigma$-definability of uncountable models of $c$-simple theories

We show that each $c$-simple theory with an additional discreteness condition has an uncountable model $\Sigma$-definable in $\mathbb{HF}(\mathbb L)$, where $\mathbb L$ is a dense linear order. From this we establish the same for all $c$-simple theories of finite signature that are submodel complete.