Context. Constraints on the internal rotation of red giants are now available thanks to asteroseismic observations. Preliminary comparisons with rotating stellar models indicate that an undetermined additional process for the internal transport of angular momentum is required in addition to purely hydrodynamic processes. Aims. We investigate how asteroseismic measurements of red giants can help us characterize the additional transport mechanism. Methods. We first determine the efficiency of the missing transport mechanism for the low-mass red giant KIC 7341231 by computing rotating models that include an additional viscosity corresponding to this process. We then discuss the change in the efficiency of this transport of angular momentum with the mass, metallicity, and evolutionary stage in the light of the corresponding viscosity determined for the more massive red giant KIC 8366239. Results. In the case of the low-mass red giant KIC 7341231, we find that the viscosity corresponding to the additional mechanism is constrained to the range νadd = 1 × 10$^3$–1.3 × 10$^4$ cm$^2$ s$^{-1}$. This constraint on the efficiency of the unknown additional transport mechanism during the post-main sequence is obtained independently of any specific assumption about the modeling of rotational effects during the pre-main sequence and the main sequence (in particular, the braking of the surface by magnetized winds and the efficiency of the internal transport of angular momentum before the post-main-sequence phase). When we assume that the additional transport mechanism is at work during the whole evolution of the star together with a solar-calibrated braking of the surface by magnetized winds, the range of νadd is reduced to 1–4 × 10$^3$ cm$^2$ s$^{-1}$-1. In addition to being sensitive to the evolutionary stage of the star, the efficiency of the unknown process for internal transport of angular momentum increases with the stellar mass.