To enhance the expressiveness of the interval temporal logic HS over branching interval models, this paper extends the language of HS by introducing quantifiers over paths. We also modify the original semantics of HS by evaluating formulas on pairs of intervals and paths, rather than on intervals alone. Our extension of HS aligns with the extension of CTL to \({\texttt {CTL}^*}\) . The resulting logic is called \(\texttt {HS}^{*}\) in this paper. We show that \(\texttt {HS}^{*}\) is strictly more expressive than HS, and we also demonstrate that \(\texttt {HS}^{*}\) is a fragment of Monadic Path Logic, a monadic second-order logic where set quantification is restricted to paths.