Stan will get your from a prior to a posterior distribution that is best supported by your data, but typically the posterior and prior distributions will not be

Stan will get your from a prior to a posterior distribution that is best supported by your data, but typically the posterior and prior distributions will not be of the same form, so there's no loop back to make. In the case that your model is extremely simple such that your posterior has a "conjugate prior" (i.e. the posterior and prior are the same family of distribution), this sort of loop back is possible. But where this is possible you have no reason at all to use Stan or MCMC since you can just update your posterior directly.