Delivery companies are offering an increasing number of time-definite services. Yet, little research has been done that explores the design of delivery networks that can support these types of services. In this paper, we explore such design problems for networks with a specified number of edges \(B > n-1\), where \(n\) is the number of nodes in the problem. We present a two-phase heuristic solution approach that first constructs a network and then improves the network via local search. For the improvement phase, we extend neighborhood structures that have proven effective for tree-structured solutions and also identify a new search neighborhood that takes advantage of specific features of subgraph solutions. We present a computational analysis of our solution approach as well as managerial insights.