  It has been observed that most manifolds in the Callahan-Hildebrand-Weeks
census of cusped hyperbolic $3$-manifolds are obtained by surgery on the
minimally twisted 5-chain link. A full classification of the exceptional
surgeries on the 5-chain link has recently been completed. In this article, we
provide a complete classification of the sets of exceptional slopes and
fillings for all cusped hyperbolic surgeries on the minimally twisted 5-chain
link, thereby describing the sets of exceptional slopes and fillings for most
hyperbolic manifolds of small complexity. The classification produces the
description of exceptional fillings for many families of one and two cusped
manifolds, and provides supporting evidence for some well-known conjectures.
One such family that appears in the classification is an infinite family of
1-cusped hyperbolic manifolds with four Seifert manifold fillings and a
toroidal filling.
