Owing to its efficiency and aptitude for a massive parallelization, the lattice Boltzmann method generally outperforms conventional solvers in terms of execution time in weakly-compressible flows. However, the authorized time-step (being inversely proportional to the speed of sound) becomes prohibitively small in the incompressible limit, so that the performance advantage over continuum-based solvers vanishes. A remedy to increase the time-step is provided by artificially tailoring the speed of sound throughout the simulation, so as to reach a fixed target Mach number much larger than the actual one. While achieving considerable speed-ups in certain flow configurations, such adaptive time-stepping comes with the flaw that the continuities of mass density and pressure cannot be fulfilled conjointly when the speed of sound is varied. Therefore, a trade-off is needed. By leaving the mass density unchanged, the conservation of mass is preserved but the pressure presents a discontinuity in the momentum equation. In contrast, a power-law rescaling of the mass density allows us to ensure the continuity of the pressure term in the momentum equation (per unit mass) but leaves the mass density locally discontinuous. This algorithm, which requires a rescaling operation of the mass density, will be called “adaptive time-stepping with correction” in the article. Interestingly, we found that this second trade-off is generally preferable. In the case of a thermal plume, whose movement is governed by the balance of buoyancy and drag forces, the correction of the mass density (to ensure the continuity of the pressure force) has a beneficial impact on the resolved velocity field. In a pulsatile channel flow (Womersley’s flow) driven by an external body force, no difference was observed between the two versions of adaptive time-stepping. On the other hand, if the pulsatile flow is established by inlet and outlet pressure conditions, the results obtained with a continuous pressure force agree much better with the analytical solution. Finally, by using adaptive time-stepping in a channel entrance flow, it was shown that the correction is compulsory for the Poiseuille flow to develop. The expected compressibility error due to the discontinuity in the mass density remains small to negligible, and the convergence rate is not notably affected compared to a simulation with a constant time step.