Mixed-mode oscillations (MMOs) are trajectories of dynamical systems in which large and small oscillations occur with a distinct gap between the two. Several types of MMOs occurring in systems with multiple timescales have been identified. The classification of these MMOs is related to the decomposition of trajectories into slow and fast segments and the types of transitions between them. An early unsuccessful attempt to locate MMOs was based upon a search for homoclinic orbits to a saddle-focus equilibrium. We revisit this search with a systematic strategy involving the computation of invariant manifolds and continuation methods. Our methods find the elusive homoclinic orbits and explain why they were difficult to find.