  The research in this paper is a continuation of the investigation of the
cardinality of the $\theta$-closed hull of subsets of spaces. This research
obtains new upper bounds of the cardinality of the $\theta$-closed hull of
subsets using cardinal functions of $\theta$-bitightness (\cite{Ca-Ko}), finite
$\theta$-bitightness (\cite{Ca-Ca-Pa-Ts}) and $\theta$-bitightness small number
(\cite{Ca-Ca-Pa-Ts}) of spaces. In the final section, examples of spaces are
presented including one that answers a question posed in \cite{Bo-Ca-Ma} and
\cite{Bo-Pa}.
