  Auckly gave two examples of irreducible integer homology spheres (one
toroidal and one hyperbolic) which are not surgery on a knot in the
three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided
infinitely many small Seifert fibered examples. In this note, we extend those
results to give infinitely many hyperbolic examples, as well as infinitely many
examples with arbitrary JSJ decomposition.
