An interesting candidate model of inflation at low energy scales comes from the MSSM, where the inflaton is a gauge-invariant flat direction and inflation occurs about a point of inflection in the potential. Such models are potentially testable at the LHC or other ground-based experiments. I will discuss a particular model of this type with a renormalizable potential, where cosmological constraints on inflation may also tie in with the smallness of the neutrino masses. Despite these advantages, however, such models require severe fine-tuning of the potential and the initial conditions for inflation may be hard to achieve. I will discuss how extending the particle content of the model to include a hybrid mechanism can help to alleviate the fine- tuning problem.