This paper deals with the development of an improved second-order theory for estimating the effective behavior of nonlinear composite dielectrics. The theory makes use of the field fluctuations in the phases of the relevant "linear comparison composite" to generate improved Maxwell-Garnett (MGA) and effective-medium (EMA) types of approximations for nonlinear media. Similar to the earlier version of the theory, the resulting MGA and EMA predictions are exact to second-order in the contrast, but—unlike the earlier version—the estimates satisfy all known bounds. In particular, the EMA estimates exhibit a nonlinearity-independent percolation threshold, and critical exponents that are consistent with recently developed bounds on these exponents. In addition, the MGA and EMA estimates are shown to yield reasonable predictions for strongly nonlinear composites with "threshold-type" nonlinearities, which are extreme cases where earlier methods have been known to sometimes fail.