Could you clarify, what did you mean by "I already have a posterior on y?"
Do you mean you have generated samples to describe the distribution of the various parameters that were fit to your data (y), in accordance with the model you defined? If this is the case, you still need to apply those parameter fits to your input data to create predictions for each observation - which you can do by "reverse engineering" your model statement, using parameter values supplied by Stan - or generating predictions in the
generated_quantities block (example here, or I can provide more detail on this if necessary, just post your original model statement). Once you have predictions, you can certainly apply any other kind of additional modeling to it, within or outside of Stan.
You might be also asking if you can "do something" with not only the point values of the parameters that came out of your first model, but also the shape of their distributions? If that's the case, I'd suggest looking into hierarchical models, which are awesome tools but come with their own set of challenges.
If you haven't seen it yet, I'd recommend John Kruschke's book Doing Bayesian Data Analysis as a great resource for this kind of thing. Chapter 9 is all about hierarchical models.