Conjectures:

For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, for some $m$ with $2<m<p$.

and

For all primes $p$ there is a prime $q<p^2$ such that $q\equiv m!\pmod {p^2}$, for some $m$ with $2<m<p^2$. (Even $2<m<p$ as pointed out in comment).

Both tested for all $p<100,000,000$.

There are reasons to believe that the conjectures are true. If $p$ is a large prime there are a lot of nonzero solutions to $n\equiv m!\!\pmod p$ and to $n\equiv m!\!\pmod {p^2}$. The probability for one of those $n$ to be a prime increase with p.

https://math.stackexchange.com/questions/1838931/conjecture-about-primes-and-the-factorial