  The silhouette of a black hole having a critical curve (an unstable bound
photon orbit) when illuminated by an optically thin accretion disk whose
emission is confined to the equatorial plane shows a distinctive central
brightness depression (the shadow) whose outer edge consists of a series of
strongly lensed, self-similar rings superimposed with the disk's direct
emission. While the size and shape of the critical curve depend only on the
background geometry, the pattern of bright and dark regions (including the size
and depth of the shadow itself) in the image is strongly influenced by the
(astro)physics of the accretion disk. This aspect makes it difficult to extract
clean and clear observational discriminators between the Kerr black hole and
other compact objects. In the presence of a second critical curve, however,
observational differences become apparent. In this work we shall consider some
spherically symmetric black hole and wormhole geometries characterized by the
presence of a second critical curve, via a uniparametric family of extensions
of the Schwarzschild metric. By assuming three toy models of geometrically thin
accretion disks, we show the presence of additional light rings in the
intermediate region between the two critical curves. The observation of such
rings could represent a compelling evidence for the existence of black hole
mimickers having multiple critical curves.
