Given a Banach space $E$ with a separable infinite-dimensional quotient, a bounded linear dense-range operator $T$ from $E$ into another Banach space $F$, and proper dense range $R$ in $E$, we provide a condition to ensure the existence of a closed subspace $E_1$ such that $E/E_1$ is infinite-codimensional and separable, $R+E_1$ is infinite-codimensional, and $T(E_1)$ is dense in $F$. As an application of this result, we deduce that if $X$ and $Y$ are proper quasicomplements in a Banach space $E$, and $X$ has a separable quotient, then $X$ contains a closed subspace $X_1$ such that $\mathrm{dim} (X/X_1)= \infty$ and $X_1+Y$ is dense in $E$. This fact generalizes a theorem of Johnson, which ensures that the assertions hold true whenever the subspace $X$ is WCG. Some refinements of these results in the case that $E$ has weak$^*$-separable dual are also given, which extend some of those obtained by Fonf, Lajara, Troyanski and Zanco in \cite{FLTZ}.
\bibitem{FLTZ} V. P. Fonf, S. Lajara, S. Troyanski and C. Zanco, \emph{Operator ranges and quasicomplemented subspaces of Banach spaces}, Studia Math. {\bf 246} (2) (2019), 203--216.