About exactness of $Hom$ functor?

Question

We know that a module $P$ is projective if and only if the functor $Hom(P,-):R-Mod \to Ab$ is exact, i.e it preserves epimorphisms: If $\alpha: M \to N$ is an epimorphism of modules then $Hom(P,\alpha):Hom(P,M)\to Hom(P,N)$ is an epimorphism of abelian groups. Now I'm trying to find some thing similar for a non-projective module $P$ but with finite projective dimension, say $pd(P)\leq d$. I know that in this case $Hom(P,-)$ is not exact, is there any condition for an epimorphism $\alpha: M \to N$ to have $Hom(P,\alpha):Hom(P,M)\to Hom(P,N)$ is an epimorphism of abelian groups?