  Locally harmonic manifolds are Riemannian manifolds in which small geodesic
spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby
parallel hypersurfaces are of constant mean curvature. Flat and rank one
symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are
non-compact harmonic manifolds, most of which are non-symmetric. Taking the
limit of an "inflating" sphere through a point $p$ in a Damek-Ricci space as
the center of the sphere runs out to infinity along a geodesic half-line
$\gamma$ starting from $p$, we get a horosphere. Similarly to spheres,
horospheres are also isoparametric hypersurfaces. In this paper, we define the
sphere-like hypersurfaces obtained by "overinflating the horospheres" by
pushing the center of the sphere beyond the point at infinity of $\gamma$ along
a virtual prolongation of $\gamma$. They give a new family of isoparametric
hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the
isoparametric hypersurfaces constructed by J. C. D\'iaz-Ramos and M.
Dom\'inguez-V\'azquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the
geometric properties of these isoparametric hypersurfaces, in particular their
homogeneity and the totally geodesic condition for their focal varieties.
