This paper introduces the concept of energy efficiency (EE) in the uplink with the capability of multi-connectivity (MC) in a multi-orbit non-terrestrial network (NTN), where user terminals (UTs) can be simultaneously served by more than one satellite to achieve higher peak throughput at reduced energy consumption. This concept also considers the service classification of the users, so that network dimensioning is performed in order to satisfy the quality of service (QoS) requirement of users. MC can increase throughput, but this entails increased power consumption at user terminal for uplink transmissions. To this end, an energy-efficient service-aware multi-connectivity (EE-SAMC) scheduling algorithm is developed in this paper to improve the EE of uplink communications. EE-SAMC uses available radio resources and propagation information to intelligently define a dynamic resource allocation pattern, that optimally routes traffic so as to reduce the energy consumption at the UT while ensuring QoS is maximized. EE-SAMC is designed based on the formulation of a non-convex combinatorial problem, it is solved in two ways involving firstly an optimization solution and secondly a heuristic approach. The effectiveness of EE-SAMC is compared with random allocation, round robin and heuristic schedulers in terms of EE, throughput and delay; EE-SAMC outperforms all schedulers.