Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map?
If it makes things easier, assume $E$ is simply connected, that is, the universal cover of $B$.
I was looking at the group of deck transformations (I don't know if they make sense in the most general settings) and it seems that sets of orbits under a fixed subgroup work. We should also be able to take union of such things, but I'm not sure under what conditions.
Any component of a preimage of a subset $A$ of $B$ is a cover of $A$, I believe (at least for open subsets). Taking the preimage of small subsets just gives you one copy of the small subset for each group element, and taking larger subsets gives you more interesting covering spaces. In fact, two preimages 'merge together' exactly when the original set contains a non-trivial loop.
For instance, a simply connected subset of the torus lifts to infinitely many disjoint copies of itself, but a big patch that loops around a meridian and touches itself lifts to disjoint copies of an 'unwinding' of itself, an infinite set that is homotopic to a line.
One problem with taking preimages of non-open subsets is that the open subset of $B$ corresponding to small neighborhoods in the non-open subset under the subspace topology may not be well-covered.