Dynamically rich nature of the high-order harmonic generation process lends itself to a variety of ways to increase photon yield and extend the harmonic cutoff frequency. We show here that high-harmonic generation from an atom confined inside an attractive potential shell can show a dramatic increase in the photon yield in certain cases. We consider an endohedrally confined hydrogen atom inside a C-60 cage as an example, and consider three distinct physical situations in which the initial state is (1) entirely confined inside the C-60, (2) partially outside, and (3) mainly localized on the cage wall. We demonstrate that when the atom-cage system starts in a state with a classical turning point outside the C-60 shell, the high-harmonic photon yield can be enhanced up to 4 orders of magnitude when compared with a free atom in the same initial state. We explain the underlying physical mechanisms in each case using fully three-dimensional quantum simulations. This gives a prime example of how directly coupling an atom to a nanostructure can alter strong field processes in atoms in interesting ways.