We study the extension of non-monotonic disjunctive logic programs with terms that represent sets of constants, called DLP(S), under the stable model semantics. This strictly increases expressive power, but keeps reasoning decidable, though cautious entailment is coNEXPTIME^NP-complete, even for data complexity. We present two new reasoning methods for DLP(S): a semantics-preserving translation of DLP(S) to logic programming with function symbols, which can take advantage of lazy grounding techniques, and a ground-and-solve approach that uses non-monotonic existential rules in the grounding stage. Our evaluation considers problems of ontological reasoning that are not in scope for traditional ASP (unless EXPTIME =ΠP2 ), and we find that our new existential-rule grounding performs well in comparison with native implementations of set terms in ASP.