A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff, and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated with one of the families. It is conjectured that there are exactly six exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea census of closed hyperbolic 3-manifolds, associated with five other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated with the seventh family, we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields.