Recently, Emerick and Reynolds introduced the ensemble smoother with multiple data assimilations (ES-MDA) for assisted history matching. Via computational examples, they demonstrated that ES-MDA provides both a better data match and a better quantification of uncertainty than is obtained with the ensemble Kalman filter (EnKF). However, like EnKF, ES-MDA can experience near ensemble collapse and results in too many extreme values of rock property fields for complex problems. These negative effects can be avoided by a judicious choice of the ES-MDA inflation factors but, prior to this work, the optimal inflation factors could only be determined by trial and error. Here, we provide two automatic procedures for choosing the inflation factor for the next data assimilation step adaptively as the history match proceeds. Both methods are motivated by knowledge of regularization procedures; the first is intuitive and heuristical; the second is motivated by existing theory on the regularization of least-squares inverse problems. We illustrate that the adaptive ES-MDA algorithms are superior to the original ES-MDA algorithm by history-matching three-phase flow production data for a complicated synthetic problem where the reservoir model parameters include the porosity, horizontal and vertical permeability fields, depths of the initial fluid contacts and the parameters of power-law permeability curves.