Thanks a million, Jan-Simon!
I think I understand it now.
The first (weak) postmark must be Piraeus.
The second postmark is Athens.
The third postmark is Kallithea, which is between Piraeus and Athens.
The fourth postmark may be Neou Falirou, which is a cross-street between Piraeus and Kallithea.
So the cover traveled from Piraeus, then to Neou Falirou, then to Kallithea and finally to Athens.
Piraeus is the port of Athens, so the cover actually traveled a
short distance (8.3 km or 5.2 miles),
which is consistent with three of the town cancels showing the same date, February 16, 1938.
One of the two Athens receiving cancels reads February 17 and the second reads February 18.