Generalizing a theorem of P. Hall on finite-by-nilpotent groups

Let $\gamma_i(G)$ and $Z_i(G)$ denote the $i$-th terms of the lower and upper central series of a group $G$, respectively. In 1956 P. Hall showed that if $\gamma_{i+1}(G)$ is finite, then the index $\vert G:Z_{2i}(G)\vert$ is finite. We prove that the same result holds under the weaker hypothesis that $\vert\gamma_{i+1}(G):\gamma_{i+1}(G)\cap Z_i(G)\vert$ is finite.