This was easier than I thought: the answer is no.

Let $f : X \to Y$ be phantom, with finite cofiber $C$. Assume without loss of generality that $C$ is connective. Then $C \to \Sigma X$ factors through $(\Sigma X)_{\geq 0}$. It follows that $\Sigma X \to (\Sigma X)_{\leq -1}$ is phantom, and in particular zero on homotopy, so that $\Sigma X$ is connective. Then it follows that $Y$ is connective as well. Then because $f$ vanishes on homology and $H_\ast C$ is finitely-generated, it follows that $H_\ast \Sigma X$ and $H_\ast Y$ are also finitely-generated. Since $\Sigma X$ and $Y$ are connective, this implies that $\Sigma X$ and $Y$ are finite. So $f = 0$.