Spekkens' toy model  can handle the measurement of a spin 1/2 along the directions $x$, $y$ and $z$. Unless I am missing something, it can't handle measurement in any direction. Would there be a known extension providing such a $\psi$-epistemic model that can handle measurement in any direction?
 Spekkens, Robert W. (2007). “Evidence for the epistemic view of quantum states: A toy theory”. In: Phys. Rev. A 75 (3), p. 032110.