Constraint acquisition (CA) is a method for learning users' concepts by representing them as a conjunction of constraints. While this approach works well for many combinatorial problems over finite domains, some applications require the acquisition of disjunctive constraints, possibly coming from logical implications or negations. In this paper, we propose the first CA algorithm tailored to the automatic inference of disjunctive constraints, named DCA. A key ingredient there, is to build upon the computation of maximal satisfiable subsets. We demonstrate experimentally that DCA is faster and more effective than traditional CA with added disjunctive constraints, even for ultra-metric constraints with up to 5 variables. We also apply DCA to precondition acquisition in software verification, where it outperforms the previous CA-based approach PreCA, being 2.5 times faster. Specifically, in our evaluation DCA infers more preconditions in just 5 minutes than PreCA does in an hour, without requiring prior knowledge about disjunction size. Our results demonstrate the potential of DCA for improving the efficiency and scalability of constraint acquisition in the disjunctive case, enabling a wide range of novel applications.