The Magellanic Clouds are a local laboratory for understanding the evolution and properties of dwarf irregular galaxies. To reveal the extended structure and interaction history of the Magellanic Clouds we have undertaken a large-scale photometric and spectroscopic study of their stellar periphery (the MAgellanic Periphery Survey, MAPS). We present first MAPS results for the Small Magellanic Cloud (SMC): Washington M, T2 + DDO51 photometry reveals metal-poor red giant branch stars in the SMC that extend to large radii (~11 kpc), are distributed nearly azimuthally symmetrically (ellipticity=0.1), and are well-fitted by an exponential profile (out to R~7.5 deg). An ~6 Gyr old, [Fe/H] -1.3 main-sequence turnoff is also evident to at least R=7.3 deg, and as far as 8.4 deg in some directions. We find evidence for a "break" population beyond ~8 radial scalelengths having a very shallow radial density profile that could be either a bound stellar halo or a population of extratidal stars. The distribution of the intermediate stellar component (3<R<7.5 deg) contrasts with that of the inner stellar component (R<3 deg), which is both more elliptical (ellipticity~0.3) and offset from the center of the intermediate component by 0.59 deg, although both components share a similar radial exponential scale length. This offset is likely due to a perspective effect because stars on the eastern side of the SMC are closer on average than stars on the western side. This mapping of its outer stellar structures indicates that the SMC is more complex than previously thought.