To improve the utilization of public transportation systems (PTSs) during off-peak hours, we present an algorithmic framework that designs PTSs with hybrid transportation units (HTUs), which can transport passengers or freight by leveraging a flexible interior. Against this background, we study a capacitated network design problem to enable cargo-hitching in existing PTSs. Specifically, we study a setting with fixed vehicle routes and timetables in which vehicles can be equipped with HTUs to enable cargo-hitching. We optimize the network design from a total cost perspective to account for normalized network design costs tied to the investment in HTUs and freight routing costs. We present an algorithmic framework that encodes some of the problem's constraints in a spatially and temporally expanded, layered graph, and solves the resulting network design problem with a price-and-branch algorithm. We apply this framework to a case study based on the subway network in the city of Munich. Our algorithm outscales commercial solvers by a factor of six and yields integer feasible solutions with a median integrality gap of less than 1.56% for all instances. We show that cargo-hitching with HTUs increases the utilization of PTSs, especially during off-peak hours, without cannibalizing passenger service level and quality. We quantify the value of hybrid transportation units (HTUs) at up to 3.2% of the total cost. Moreover, we present a sensitivity analysis that indicates that cargo-hitching is worthwhile if truck-based transport occurs at an externality cost of more than 1.5 EUR per vehicle and kilometer and loading and unloading costs of less than 2.0 EUR per passenger equivalent.