Characterizations of $\beta$-connectedness and $\Cal S$-connectedness of topological spaces are investigated. Further results concerning preservation of these connectedness-like properties under surjections are obtained. The paper completes our previous study [Z. Duszy\'nski, {t On some concepts of weak connectedness of topological spaces}, Acta Math. Hungar. {\bf 110} (2006), 81-90].