<p>Few-body scattering and loosely bound states in the vicinity of an isolated narrow Feshbach resonance are well described by a conceptually simple contact interaction two-channel model, where an open atomic continuum channel is coupled to a closed molecular channel. However, real systems are often characterized by the multiplicity of both continuum and molecular channels. Here we develop a systematic framework to deal with multi-channel problems within a contact interaction approximation. We demonstrate the differences and similarities of adding a molecular or atomic channel to the two-channel model. By means of direct comparison, we show that while an additional atomic channel makes loosely bound states shallower, an additional molecular channel makes them more deeply bound. This approach facilitates the study of the influence of multi-channel environments on few-body physics in a systematic manner by gradually increasing the level of complexity. Furthermore, we account for real atomic systems whose understanding can benefit from this study.</p>