Schubert came extremely close to discovering the JSJ-decomposition of 3-manifolds in his paper "Knoten und Vollringe" (1953). With a little more work, one could turn Schubert's paper into something equivalent to the JSJ-decomposition applied to knot and link complements in $S^3$. That would have allowed people to conjecture the JSJ-decomposition for 3-manifolds, around 20 years earlier than it was.
It's interesting to speculate whether or not the connection between 3-manifold theory and hyperbolic geometry would have been made much earlier, as some of the ingredients were already in place -- Seifert-Weber space, and the Gieseking manifold, but I do not think people knew finite-volume hyperbolic manifolds to be atoroidal until much later.