In his book "Stable Homotopy and Generalized Homology" Adams takes a ring spectrum E (let's say E_infinity for simplicity) such that $E_*E$ is flat over $\pi_*E$. He claims that the spectrum level map $E\wedge E \wedge E \wedge X\to E\wedge E\wedge X$ given by taking the product on the two inner copies of $E$ induces a map $E_*(E)\otimes_{\pi_*E}E_*X\to [S, E\wedge E\wedge X]$.
I am not sure how to see what I assume is some map (isomorphism?) from the domain of that map to $\pi_*(E\wedge E\wedge E\wedge X)$.
My guess is that this is some simply trick using the flatness assumption, but it is not obvious to me.
As a more general question, is there a good resource for when we can turn homotopy groups of smash products into tensor products of homotopy groups over some ring?
If I am not mistaken there is an extra copy of $E$ everywhere. The map should be $E \wedge E \ \wedge X \rightarrow E \wedge X$, and the map you are asking about is $E_*(E) \otimes_{\pi_*(E)} E_*(X) \rightarrow \pi_*(E \wedge X)=E_*(X)$. This is otherwise known as the action of homology (co?)operations on homology. Let's proceed with this new map:
When $E_*(X)$ is flat over $\pi_*(E)$, the Kunneth spectral sequence for $E$ turns into the Kunneth isomorphism $E_*(X \wedge Y) \cong E_*(X) \otimes_{\pi_*(E)} E_*(Y)$ because flatness makes $\operatorname{Tor}^{\pi_*(E)}_{>0}(E_*(X),-)$ vanish.
Now applying $\pi_*$ to the map $E \wedge E \wedge X \rightarrow E \wedge X$, we suggestively bracket the domain as $E \wedge (E \wedge X)$ and recalling that the $E$-homology of a spectrum is the same as the homotopy groups of the smash product with $E$, so we see the left side is $E_*(E \wedge X)$. Now the flatness assumption of $E_*(E)$ yields the Kunneth isomorphism with $E_*(E) \otimes_{\pi_*(E)} E_*(X)$, and the right hand side needs no adjusting.