This paper presents a refinement of a result by Conidis, who proved that there is a real $X$ of effective packing dimension $0\lt \alpha\lt 1$ which cannot compute any real of effective packing dimension $1$. The original construction was carried out below $\emptyset''$, and this paper’s result is an improvement in the effectiveness of the argument, constructing such an $X$ by a limit-computable approximation to get $X\leq_{T}\emptyset'$.