When designing an assortment, a fundamental trade-off exists between estimating consumer behavior more precisely and solving the corresponding optimization problem efficiently. For instance, a mixture of logit model can closely approximate any random-utility choice model; however, it is not suitable for optimization when the number of customer segments and the number of products considered are large. As many companies are using big data and marketing analytics toward microsegmenting consumers, a choice model that is amenable in the problem size becomes necessary for assortment design. In this work, we provide a new approach to approximate any random-utility choice model by characterizing the estimation errors in a classical exogenous demand model and significantly improving its performance with a rescaling method. We show that the resulting approximation is exact for Multinomial Logit (MNL). If, however, the underlying true choice model is not MNL, we show numerically that the approximation under our so-called rescaled two-attempt model outperforms the widely used MNL approximation, and provides performance close to the Markov chain approximation (in some cases, it performs better than the Markov chain approximation). Our proposed approximation can be used to solve a general assortment optimization problem with a variety of (linear) real-world constraints. In contrast to the more direct Mixed Integer Optimization (MIO) approach that utilizes Latent Class Multinomial Logit (LC-MNL), whose running time increases exponentially in the number of mixtures, our approximation yields an alternative MIO formulation whose empirical running time is independent of the number of mixtures.