We give several new bounds for the cardinality of a Hausdorff topological space X involving the weak Lindelöf degree ωL(X). In particular, we show that if X is extremally disconnected, then |X| ≤ 2ωL(X)πχ(X)ψ(X), and if X is additionally power homogeneous, then |X| ≤2ωL(X)πχ(X). We also prove that if X is a star-DCCC space with a Gδ -diagonal of rank 3, then |X| ≤ 2ℵ0 ; and if X is any normal star-DCCC space with a Gδ -diagonal of rank 2, then |X| ≤2ℵ0. Several improvements of results in [10] are also given. We show that if X is locally compact, then |X| ≤ ωL(X)ψ(X) and that |X| ≤ ωL(X)t(X) if X is additionally power homogeneous. We also prove that |X| ≤ 2ψc(X)t(X) ωL(X) for any space with a π-base whose elements have compact closures and that the stronger inequality |X| ≤ωL(X)ψc(X)t(X) is true when X is locally H-closed or locally Lindelöf. [ABSTRACT FROM AUTHOR]