I don't care if he signs a contract with a private agency, if it relates to his job as a state employee, then it should be public information.
If I write a letter to MSU, that is a public document. If I write a letter to the Head Coach of the MSU football team, that should also be a public document. If I write a letter to Dan Mullen as a private citizen, that is a different matter: he can make it public if he wants to, but you cannot force him.
But, as I said, I don't know the details of the Mississippi FOIA laws.