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Large diffuse galaxies are hard to find, but understanding the environments where they live, their numbers, and ultimately their origins, is of intense interest and importance for galaxy formation and evolution. Using Subaru's Hyper Suprime-Cam Strategic Survey Program, we perform a systematic search for low surface brightness galaxies and present novel and effective methods for detecting and modeling them. As a case study, we surveyed 922 Milky Way analogs in the nearby Universe ($0.01 < z < 0.04$) and build a large sample of satellite galaxies that are outliers in the mass-size relation. These ``ultra-puffy'' galaxies (UPGs), defined to be $1.5σ$ above the average mass-size relation, represent the tail of the satellite size distribution. We find that each MW analog hosts $N_{\rm UPG} = 0.31\pm 0.05$ ultra-puffy galaxies on average, which is consistent with but slightly lower than the observed abundance at this halo mass in the Local Volume. We also construct a sample of ultra-diffuse galaxies (UDGs) in MW analogs and find an abundance of $N_{\rm UDG} = 0.44\pm0.05$ per host. With literature results, we confirm that the UDG abundance scales with the host halo mass following a sublinear power law. We argue that our definition for ultra-puffy galaxies, which is based on the mass-size relation, is more physically-motivated than the common definition of ultra-diffuse galaxies, which depends on surface brightness and size cuts and thus yields different surface mass density cuts for quenched and star-forming galaxies.