Neural architecture search has proven to be highly effective in the design of efficient convolutional neural networks that are better suited for mobile deployment than hand-designed networks. Hypothesizing that neural architecture search holds great potential for human pose estimation, we explore the application of neuroevolution, a form of neural architecture search inspired by biological evolution, in the design of 2D human pose networks for the first time. Additionally, we propose a new weight transfer scheme that enables us to accelerate neuroevolution in a flexible manner. Our method produces network designs that are more efficient and more accurate than state-of-the-art hand-designed networks. In fact, the generated networks process images at higher resolutions using less computation than previous hand-designed networks at lower resolutions, allowing us to push the boundaries of 2D human pose estimation. Our base network designed via neuroevolution, which we refer to as EvoPose2D-S, achieves comparable accuracy to SimpleBaseline while being 50% faster and 12.7x smaller in terms of file size. Our largest network, EvoPose2D-L, achieves new state-of-the-art accuracy on the Microsoft COCO Keypoints benchmark, is 4.3x smaller than its nearest competitor, and has similar inference speed. The code is publicly available at https://github.com/wmcnally/evopose2d.