  The set of equivalence classes of cobounded actions of a group G on different
hyperbolic metric spaces carries a natural partial order. Following
Abbott--Balasubramanya--Osin, the group G is H--accessible if the resulting
poset has a largest element. In this paper, we prove that every non-geometric
3--manifold has a finite cover with H--inaccessible fundamental group and give
conditions under which the fundamental group of the original manifold is
H--inaccessible. We also prove that every Croke--Kleiner admissible group (a
class of graphs of groups that generalizes fundamental groups of 3--dimensional
graph manifolds) has a finite index subgroup that is H--inaccessible.
