Despite the fact that medical responses are crucial for saving precious lives during any humanitarian crisis (e.g., the COVID-19 pandemic), healthcare infrastructure in many communities are partially covered or are not covered yet. In order to strengthen the health system response to such crisis, especially in low-to middle-income communities, this paper extends a novel multi-objective model for designing a health service network under uncertainty which simultaneously considers efficiency, social responsibility, and network cost. For efficiency, a modified data envelopment analysis model is introduced and inserted into the proposed model to decrease the inefficiency of healthcare facilities belonging to the different tiers of the health system. For social responsibility, two measures of job creation and balanced development are incorporated into the extended model. This is not only considered to cope with the increased numbers of patients and disaster victims to healthcare facilities but also to deal with the challenge of the economy and the livelihoods of people during the crisis. Moreover, a novel mixed possibilistic-flexible robust programming (MPFRP) approach is developed to protect the considered network against uncertainty. To show the applicability of the extended model, a real-world case study is presented. The results reveal that contrary to fuzzy programming models, the MPFRP performs well in terms of social responsibility (72%), cost (8%), and efficiency (28%) and is able to make a trade-off between these three measures. In this study, the resilience level of the designed network is not addressed while disregarding any short-term stoppage owing to internal or external sources of disruption in designing may bring about a considerable loss.