On the one hand, Unmanned Aircraft Systems (UASs) have experienced great applicability surge in the recent years, arising as a promising technology with a wide field of use. On the other hand, healthcare, a critical system in modern society, is subject to a heavy and unexpected pressure in the case of situations such as the COVID-19 pandemic. This article aims to leverage the flexibility of UASs as complementary support for healthcare logistic systems when under high-stress conditions, via quick deployment of an air delivery network. We have defined a logistics network model and created three scenarios based on the model and current needs in Valencia (Spain). Flight tests have been performed in these scenarios, which include urban areas and controlled airspace. Operations complied with requirements derived from the application of Specific Operations Risk Assessment (SORA) methodology, recently adopted by the European Aviation Safety Agency (EASA). Flights were successful, being able to swiftly deliver medical goods without requiring any dedicated infrastructure. However, a moderate number of contingencies took place during the tests, mainly related to control link quality and Air Traffic Management (ATM) integration, forcing the use of dedicated procedures to cope with them. Although additional development is required to ensure the safety of large-scale automated operations, the use of UASs as part of logistic networks is a feasible means to support existing structures, especially in situations in dire need.