We know that every universal covering space is simply connected. The converse is trivially true, every simply connected space is a covering space of itself. But I'm wondering what simply connected spaces, specifically manifolds, are covering spaces of another (connected) space other than itself.
I found this paper that shows that there exist certain contractible open simply connected manifolds that aren't non-trivial covering spaces of manifolds. If we restrict to closed simply connected manifolds, can we also find such examples, or can there always be a different manifold that it covers?