In typical expositions of renormalization of composite operators, one needs such renormalization to tame the divergences of some matrix elements such as (say in an interacting scalar theory)

\[\langle \alpha|\phi^2(x)|\beta\rangle\]

One renormalizes $\phi^2$ by introducing the renormalized composite operator

\[(\phi^2)_R=Z_{(\phi^2)}\phi^2.\]

My question is, doesn't the above relation imply $Z_{(\phi^2)}=Z^2_{\phi}$, where $Z_{\phi}$ is the field strength renormalization of $\phi$? That is, once the elementary field is renormalized, how come one can still have the freedom to renormalize the composite operator?

Useful scholarpedia introduction on the subject(still scratching my head reading it): http://www.scholarpedia.org/article/Local_operator#Definition_of_normal_products_and_operator-mixing