We revise the simple and yet powerful approach of the Double Adiabatic Approximation (DAA) to study positive muon embedding sites in condensed matter. This method introduces a tractable and reasonably accurate description of the quantum nature of the muon in Density Functional Theory (DFT) simulations aiming at identifying muon sites. Moreover, this approach overcomes the limitations of the harmonic approximation in terms of both computational efforts and accuracy. Notably, it also improves treatment of muon diffusion. We use the DAA and Nudged Elastic Band calculations to determine the muon sites and the activation energies between equivalent sites in MnSi and copper. In metallic Cu, the proposed phonon-assisted mechanism for the low-temperature muon diffusion in the lattice is confirmed.