jats:titleAbstract</jats:title> jats:pIn this two-part paper we review, and then develop, the assessment of the hole argument for general relativity.</jats:p> jats:pThe review (in Part I) discussed how to compare points in isomorphic spacetimes, i.e. models of the theory. This second Part proposes a framework for making comparisons of non-isomorphic spacetimes. It combines two ideas we discussed in Part I—the philosophical idea of jats:italiccounterparts</jats:italic>, and the idea of threading points between spacetimes other than by isomorphism—with the mathematics of fibre bundles.</jats:p> jats:pWe first recall the ideas from Part I (Section 1). Then in Section 2 and an Appendix, we define a fibre bundle whose fibres are isomorphic copies of a given spacetime or model, and discuss connections on this fibre bundle. This material proceeds on analogy with field-space formulations of gauge theories. Finally, in Section 3, we show how this fibre bundle gives natural expressions of the philosophical ideas of counterparts, and of threading.</jats:p>