The discovery probability of long-period comets (LPCs) passing near the Sun is highest during their first passage and then declines, or fades, during subsequent return passages. Comet fading is largely attributed to devolatilization and fragmentation via thermal processing within 2--3 au of the Sun (1 au being the Earth-Sun distance). Here our numerical simulations show that comet observing campaigns miss vast numbers of LPCs making returning passages through the Saturn region (near 10 au) because these comets fade during prior, even more distant passages exterior to Saturn and thus elude detection. Consequently, comet properties significantly evolve at solar distances much larger than previously considered, and this offers new insights into the physical and dynamical properties of LPCs, both near and far from Earth.