  We show that any co-oriented closed contact manifold of dimension at least
five admits a contact form such that the contact volume is arbitrarily small
but the Reeb flow admits a global hypersurface of section with the property
that the minimal period on the boundary of the hypersurface and the first
return time in the interior of the hypersurface are bounded below. An immediate
consequence of this statement is that every co-oriented contact structure on
any closed manifold admits a contact form with arbitrarily large systolic
ratio. This generalizes the result of Abbondandolo et al. in dimension three to
higher dimensions. The proof the main result is inductive and uses the result
of Abbondandolo et al. on large systolic ratio in dimension three in its basis
step. The essential construction in the proof relies on the Giroux
correspondence in higher dimensions.
