New physics beyond General Relativity can modify image features of black holes and horizonless spacetimes and increase the separation between photon rings. This motivates us to explore synthetic images consisting of two thin rings. Our synthetic images are parameterized by the separation as well as the relative flux density of the two rings. We perform fits to the visibility amplitude and analyze closure quantities. The current Event Horizon Telescope array cannot detect the presence of a second ring in the region of parameters motivated by particular new-physics cases. We show that this can be improved in three ways: first, if the array is upgraded with Earth-based telescopes with sufficiently high sensitivity, second, if the array is upgraded with a space-based station and third, if super-resolution techniques are used for the data obtained by the array.