We investigate the computational complexity of Minimum Coverage in Multi-Interface Networks (Min CMI), which has applications in the field of Internet of Things (IoT). In this problem, we are given an undirected graph where each vertex represents a device (e.g., smartphones, environmental sensors, smart speakers) capable of activating multiple interfaces (e.g., Bluetooth, Wi-Fi, 4G/5G) to establish connections with one another. The objective is to ensure the connectivity requirement of the network while minimizing its total energy consumption by activating only the necessary interfaces. Contributing to the computational complexity landscape of Min CMI, we unify and extend previously known algorithmic results and demonstrate their limits. In particular, we analyze two scenarios based on the number of different interfaces across the network. When this number is unbounded, we show that the problem remains fixed-parameter intractable with respect to the number p of active interfaces per node within highly restricted graph classes such as stars and cliques. Additionally, we show that even in the case that p is a small constant, strong structural parameters like vertex cover number, feedback edge number, or distance to clique do not help to obtain fixed-parameter tractability. On the positive side, we provide a polynomial-time algorithm for cographs when  \(p=2\) . When the number of different interfaces is bounded by a constant, the problem, while still NP-hard, allows for more tractable special cases, including sparse graphs with small separators (i.e., low treewidth) and dense graphs.