What's the situation where minor version upgrades are giving you trouble?
Asking because it's not supposed to be troublesome. In theory (!), minor version upgrades don't require any change to the on-disk data, so you should be able to just upgrade your PG binaries then restart the database.
Ah, looks like psql seems to have switched to actually using minor and major versions properly during the last two years - I remember the dance from the 9.x versions and postgres-upgrade [1].
In any case it's way more straightforward with mysql.
Asking because it's not supposed to be troublesome. In theory (!), minor version upgrades don't require any change to the on-disk data, so you should be able to just upgrade your PG binaries then restart the database.