Converse to Composition of Finite Homomorphisms is Finite

Question

Let $A,B,C$ be rings. I know if $A \to B \to C$ where $A \to B$ and $B \to C$ are finite, then $A \to C$ is finite (i.e. as modules). My question is, if we know $A \to C$ is finite, does it follow that $A \to B$ or $B \to C$ is finite. Specifically, what if in this case $A$ is Noetherian and $B = A[x]$ is also Noetherian by Hilbert's basis theorem?