This paper describes a fully automatic pipeline for finding an intrinsic map between two non-isometric, genus zero surfaces. Our approach is based on the observation that efficient methods exist to search for nearly isometric maps (e.g., Möbius Voting or Heat Kernel Maps), but no single solution found with these methods provides low-distortion everywhere for pairs of surfaces differing by large deformations. To address this problem, we suggest using a weighted combination of these maps to produce a ``blended map.'' This approach enables algorithms that leverage efficient search procedures, yet can provide the flexibility to handle large deformations. The main challenges of this approach lie in finding a set of candidate maps mi and their associated blending weights bi(p) for every point p on the surface. We address these challenges specifically for conformal maps by making the following contributions. First, we provide a way to blend maps, defining the image of p as the weighted geodesic centroid of mi(p). Second, we provide a definition for smooth blending weights at every point p that are proportional to the area preservation of mi at p. Third, we solve a global optimization problem that selects candidate maps based both on their area preservation and consistency with other selected maps. During experiments with these methods, we find that our algorithm produces blended maps that align semantic features better than alternative approaches over a variety of data sets.