In this paper, we prove that a space $X$ has a $\sigma$-locally countable Lindelöf $sn$-network if and only if $X$ is a compact-covering compact $msss$-image of a locally separable metric space, if and only if $X$ is a sequentially-quotient $\pi$ and $msss$-image of a locally separable metric space, where ``compact-covering'' (or ``sequentially-quotient'') can not be replaced by ``sequence-covering''. As an application, we give a new characterizations of spaces with $\sigma$-locally countable Lindelöf weak bases.