This answer by Farmer S to another question, which Farmer S linked in a comment to this question, proves that if $\alpha$ is the least ordinal that is the height of a transitive model of $\text{ZFC}+0^\#$ and $\beta$ is the least admissible ordinal greater than $\alpha$, then there is such a model coded by a set in $L_{\beta+1}$. The proof seems to work for every theory extending $\text{ZFC}+0^\#$.