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The effects of hyperfine structure on ultracold molecular collisions in external fields are largely unexplored due to major computational challenges associated with rapidly proliferating hyperfine and rotational channels coupled by highly anisotropic intermolecular interactions. We explore a new basis set for incorporating the effects of hyperfine structure and external magnetic fields in quantum scattering calculations on ultracold molecular collisions. The basis is composed of direct products of the eigenfunctions of the total {\it rotational} angular momentum (TRAM) of the collision complex Jr and the electron/nuclear spin basis functions of the collision partners. The separation of the rotational and spin degrees of freedom ensures rigorous conservation of Jr even in the presence of external magnetic fields and isotropic hyperfine interactions. The resulting block-diagonal structure of the scattering Hamiltonian enables coupled-channel calculations on highly anisotropic atom-molecule and molecule-molecule collisions to be performed independently for each value of Jr, with an added advantage of eliminating the unphysical states present in the total angular momentum representation. We illustrate the efficiency of the TRAM basis by calculating state-to-state cross sections for ultracold He + YbF collisions in a magnetic field. The size of the TRAM basis required to reach numerical convergence is 8 times smaller than that of the uncoupled basis used previously, providing a computational gain of three orders of magnitude. The TRAM basis is therefore well suited for rigorous quantum scattering calculations on ultracold molecular collisions in the presence of hyperfine interactions and external magnetic fields.