New ekpyrotic cosmology is an alternative scenario of early universe cosmology in which the universe existed before the big bang. The simplest model relies on two scalar fields, whose entropy perturbation leads to a scale-invariant spectrum of density fluctuations. The ekpyrotic solution has a tachyonic instability along the entropy field direction which, a priori, appears to require fine-tuning of the initial conditions. In this paper, we show that these can be achieved naturally by adding a small positive mass term for the tachyonic field and coupling to light fermions. Then, for a wide range of initial conditions, the tachyonic field gets stabilized with the appropriate values well before the onset of the ekpyrotic phase. Furthermore, we show that ekpyrotic theory is successful in solving the flatness, horizon and homogeneity problems of standard big bang cosmology. Motivated by the analysis of the tachyonic instability, we propose a simplification of the model in terms of new field variables. Instead of requiring two exponential potentials, one for each scalar field, it suffices to consider a single nearly exponential potential for one of the fields and a tachyonic mass term along the orthogonal direction in field space. All other terms in the potential are essentially arbitrary. This greatly widens the class of ekpyrotic potentials and allows substantial freedom in determining the spectral index and possible non-Gaussianity. We present a generalized expression for the spectral index which is easily consistent with the observed red tilt. We also argue that for a wide range of potentials non-Gaussianity can be substantial, within the reach of current observations.