One of the main aims of the methods developed for reasoning under inconsistency, in particular paraconsistent inference, is to derive informative conclusions from inconsistent bases. In this paper, we introduce an approach based on inconsistency measurement for defining non-monotonic paraconsistent consequence relations. The main idea consists in adapting properties of classical reasoning under consistency to inconsistent propositional bases by involving inconsistency measures (IM). We first exhibit interesting properties of our consequence relations. We then study situations where they bring about consequences that are always jointly consistent. In particular, we introduce a property of inconsistency measures that guarantees the consistency of the set of all entailed formulas. We also show that this property leads to several interesting properties of our IM-based consequence relations. Finally, we discuss relationships between our framework and well-known consequence relations that are based on maximal consistent subsets. In this setting, we establish direct connections between the latter and properties of inconsistency measures.