So many years without right answer... Of course, you can!

Just stop PG, make copy of your cluster data directory (PGDATA) with thoroughly saved permissions and change in your PG`s postgresql.conf "data_directory" parameter pointing to the new location, start PG.

I.e.

/etc/postgresql/11/main/postgresql.conf

data_directory = '/mnt/other_storage/new_cluster_location'

It was tested many times under Debian and Ubuntu environments without any problems. Just works as it expected: fast and reliable (PG versions 9-16).

data_directory in pg_catalog->pg_settings changes automatically after server restarts.