Expectation propagation might be the proper way to do but it is not available in PyMC yet: (not sure if it available at all in other PPLs). The challenge is that you want to fit the new time series but also update the old posterior you have.
And alternative is to construct a good VI approximation, and train it with mini-batch. Then you can treat new time series as new batches.
All in all: https://twitter.com/junpenglao/status/1453298183132651522?s=20