The answer is "No" in general, and the special class of rings whose right modules all have projective covers is the class of right perfect rings.
One form of Nakayama's lemma is that J(R)M is a superfluous submodule of M when M is a finitely-generated module over R. This definition can be generalized to an arbitrary abelian category C. An essential extension is a monomorphism u : M → E such that for every non-zero subobject s : N → E, the fibre product N ×E M ≠ 0.
If X has an injective hull Y, then Y is the largest essential extension of X (Porst 1981, Introduction (v)).
But the largest essential extension may not be an injective hull.
Indeed, in the category of T1 spaces and continuous maps, every object has a unique largest essential extension, but no space with more than one element has an injective hull (Hoffmann 1981).