<p>Naked singularity spacetimes in general relativity can be naturally divided into two regions: (i) the region exterior to the past light cone of the singularity, that extends all the way to (an incomplete) future null infinity and ensures the nakedness property (what we will call <i>exterior-naked singularity regions</i>), (ii) the interior fill-in that ensures that the singularity arises from regular initial data. The problem of constructing naked singularities has been resolved for the spherically symmetric Einstein-scalar field system by Christodoulou (Ann Math(2) 140(3):607–653, 1994), but his construction, based on a continuously self-similar ansatz, requires that both the exterior and the interior regions are mildly irregular on the past cone of the singularity. On the other hand, numerical works suggest that there exist naked singularity spacetimes with discrete self-similarity arising from smooth initial data. In this paper, we construct <i>exterior-naked singularity regions</i> with discretely self-similar profiles which are smooth on the past cone of the singularity. We show that the scalar field remains uniformly bounded, but the singularity is characterized by the infinite oscillations of the scalar field and the mass aspect ratio. Our examples require however that the mass aspect ratio is uniformly small, and thus the solutions are distinct from the exterior regions of the numerical examples.</p>