Mapping the elastic and anelastic structure of the Earth's mantle is crucial for understanding the temperature, composition and dynamics of our planet. In the past quarter century, global tomography based on ray theory and first-order perturbation methods has imaged long-wavelength elastic velocity heterogeneities of the Earth's mantle. However, the approximate techniques upon which global tomographers have traditionally relied become inadequate when dealing with crustal structure, as well as short-wavelength or large amplitude mantle heterogeneity. The spectral element method, on the other hand, permits accurate calculation of wave propagation through highly heterogeneous structures, and is computationally economical when coupled with a normal mode solution and applied to a restricted region of the Earth such as the upper mantle (SEM). Importantly, SEM allows a dramatic improvement in accounting for the effects of crustal structure. Here, we develop and apply a new hybrid method of tomography, which allows us to leverage the accuracy of SEM to model fundamental and higher-mode long period (>60 s) waveforms. We then present the first global model of upper-mantle velocity and radial anisotropy developed using SEM. Our model, SEMum, confirms that the long-wavelength mantle structure imaged using approximate semi-analytic techniques is robust and representative of the Earth's true structure. Furthermore, it reveals structures in the upper mantle that were not clearly seen in previous global tomographic models. We show that SEMum favourably compares to and rivals the resolving power of continental-scale studies. This new hybrid approach to tomography can be applied to a larger and higher-frequency data set in order to gain new insights into the structure of the lower mantle and more robustly map seismic structure at the regional and smaller scales.