We present the topology of the coronal magnetic field continuously between 1 and 6 Rs, as quantitatively inferred for 14 unique eclipse coronae that span almost two complete solar cycles. The magnetic field topology was traced automatically in processed white-light eclipse images using the Rolling Hough Transform (RHT). We find that the direction of the coronal magnetic field does not become radial until at least 3 Rs, which challenges the common assumption of a radial field at 2.5 Rs in source surface driven modeling (i.e. PFSS). In addition, we find an abundance of field lines which extend continuously from the solar surface out to several solar radii at all latitudes, regardless of the presence of coronal holes, and a high variance in radiality between 1.5 and 3 Rs at different latitudes and phases of the solar cycle. The most non-radial coronal field topologies are found to occur above regions with weaker magnetic field strengths in the photosphere, while stronger photospheric fields are associated with highly radial field lines in the corona. These results have implications for testing and constraining coronal magnetic field models, and for linking in situ solar wind measurements to their sources at the Sun.