The present work investigates the spontaneous acceleration of premixed flames in micro-channels in the process of deflagration-to-detonation transition. It has recently been shown experimentally [Wu et al., Proc. Combust. Inst. 31 (2007) 2429], computationally [Valiev et al., Phys. Rev. E 80 (2009) 036317] and analytically [Bychkov et al., Phys. Rev. E 81 (2010) 026309] that the flame acceleration undergoes a number of stages from an initial exponential regime to quasi-steady fast deflagration. The present work focuses on the final saturation stages in the process of flame acceleration, during which the flame propagates with supersonic velocity with respect to the tube wall. It is shown that an intermediate stage with quasi-steady velocity noticeably below the Chapman-Jouguet deflagration speed may be observed during the acceleration process. The intermediate stage is followed by additional flame acceleration and subsequent saturation to the Chapman-Jouguet deflagration regime. We explain the intermediate stage by the combined effects of gas pre-compression ahead of the flame front and the hydraulic resistance. We estimate the first quasi-steady saturation velocity theoretically and compare it with the numerical results. Numerical simulation shows that, in agreement with the theoretical prediction, heating due to viscous stress at the wall is minor before the flame reaches the first quasi-steady stage and is prevailing afterwards. The additional acceleration is related to viscous heating at the channel walls, being of key importance at the final stages. The possibility of explosion triggering is also demonstrated.