The structure of the outer disk of planetesimals in today's solar system was shaped largely by the dynamical history of Neptune. In the Classical Region of the Kuiper Belt, there exist overlying populations of high-inclination, dynami- cally hot Kuiper Belt objects (KBOs) and low-inclination cold objects, each with uniquely defined properties. The characteristics of these populations can provide insight into Neptune's past. To date, there has been no successful simulation of Neptune's history that scatters the hot KBOs to their observed locations today and preserves the cold population without secularly exciting the objects to eccentricities beyond that of the observational data. However, it has been suggested that if, in Neptune's migration from its formation location interior to the present-day planet, it experienced a period of high eccentricity, it could scatter some hot objects into the classical region without exciting the cold KBOs to high eccentricities. We explore the eccentricity/semimajor axis parameter space of Neptune in an attempt to find a model that matches the observed data. In our simulations, we provide Neptune with an initial eccentricity and then damp it over relevant timescales to the fairly circular orbit it exhibits today. While the examples we run provide histories of Neptune that support the conservative constraints placed on the hot and cold KBOs, there appears to be no point in our initial parameter space capable of producing a Kuiper Belt that accurately matches that of our current solar system. However, in testing more extreme initial conditions, the resulting classical objects seemed to match their respective observed populations, verifying this model as a possible history of Neptune.