We consider mixed-integer linear chance-constrained problems for which the random vector that parameterizes the feasible region has finite support. Our key objective is to improve branch-and-bound or -cut approaches by introducing new types of valid inequalities that improve the dual bounds and, by this, the overall performance of such methods. We introduce so-called primal-dual as well as covering valid inequalities. By re-scaling the latter inequalities, we obtain so-called multi-disjunctive valid inequalities, which generalize known inequalities from the literature. We provide theoretical results regarding dominance relations, closure properties, and hardness of the separation problems. Given these insights, we propose heuristic separation procedures and present extensive numerical results showing the effectiveness of our method in comparison to state-of-the-art inequalities from the literature.