Many models of high energy physics possess metastable vacua. It is conceivable that the universe can get trapped in such a false vacuum, irrespective of its origin and prior history, at an earlier stage during its evolution. The ensuing false vacuum inflation results in a cold and empty universe and has a generic graceful exit problem. We show that an inflection point inflation along the flat directions of the Minimal Supersymmetric Standard Model (MSSM) can resolve this graceful exit problem by inflating the bubble which nucleate out of a false vacuum. The important point is that the initial condition for an MSSM inflation can be naturally realized, due to an attractor behavior towards the inflection point. We investigate these issues in detail and also present an example where metastable vacua, hence the false vacuum inflation, can happen within the MSSM.