We use the new Type Ia supernovae discovered by the Sloan Digital Sky Survey-II supernova survey, together with additional supernova data sets as well as observations of the cosmic microwave background and baryon acoustic oscillations to constrain cosmological models. This complements the standard cosmology analysis presented by Kessler et al. in that we discuss and rank a number of the most popular nonstandard cosmology scenarios. When this combined data set is analyzed using the MLCS2k2 light-curve fitter, we find that more exotic models for cosmic acceleration provide a better fit to the data than the ΛCDM model. For example, the flat Dvali-Gabadadze-Porrati model is ranked higher by our information-criteria (IC) tests than the standard model with a flat universe and a cosmological constant. When the supernova data set is instead analyzed using the SALT-II light-curve fitter, the standard cosmological-constant model fares best. This investigation of how sensitive cosmological model selection is to assumptions about, and within, the light-curve fitters thereby highlights the need for an improved understanding of these unresolved systematic effects. Our investigation also includes inhomogeneous Lemaître-Tolman-Bondi (LTB) models. While our LTB models can be made to fit the supernova data as well as any other model, the extra parameters they require are not supported by our IC analysis. Finally, we explore more model-independent ways to investigate the cosmic expansion based on this new data set.