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We continue the study of almost-$\omega$-resolvable spaces beginning in A. Tamariz-Mascar’ua, H. Villegas-Rodr’ıguez, , Comment. Math. Univ. Carolin. (2002), no. 4, 687–705. We prove in ZFC: (1) every crowded $T_0$ space with countable tightness and every $T_1$ space with $\pi$-weight $\le {\aleph }_1$ is hereditarily almost-$\omega$-resolvable, (2) every crowded paracompact $T_2$ space which is the closed preimage of a crowded Fréchet $T_2$ space in such a way that the crowded part of each fiber is $\omega$-resolvable, has this property too, and (3)...