Methods of procuring a core sample may involve engaging an earth formation with a cutting structure of a coring bit. A core sample may be received within a receptacle connected to the coring bit, the receptacle being lined with a sponge material. A space of about 1 mm or less may be maintained between the core sample and the sponge material. Coring tools may include a coring bit comprising an inner gage and an outer gage and a sponge material positioned to at least partially surround a core sample cut by the coring bit. A radial distance between an inner surface of the sponge material and the inner gage of the coring bit may be about 1 mm or less. A distance between a center of curvature of the inner gage and a center of curvature of the outer gage may be about 0.3 mm or less.