That is simply an aition explaining the name of Kefalonia and reinforcing its cultural connections with Athens, Kefalos helped Amphitryon of Mycenae in a war against the Taphians and Teleboans. He was awarded with the island of Samos, which thereafter came to be known as Kefalonia. The people who lived on Kefalonia and nearby islands came to be known as Kefalonians.

Kefalos eventually married again, choosing a daughter of Minyas to be his wife. This woman (named Clymene, according to some sources) had a son with him named Arcesius. In another version, Kefalos consulted an oracle, and the oracle told him that if he wished to have a son, he should mate with the first female being he saw. Kefalos then encountered a she-bear, and mated with the animal. She then transformed into a human woman and gave birth to Arcesius. Arcesius succeeded Cephalus as ruler of his Cephallenian realm. This Arcesius was sometimes said to be the grandfather of Odysseus. In another version he had four sons after which four cities were named: Same, Crane, Pali, Pronnoi. These are the cities who later became the four city-states of Kefalonia. Nevertheless, Kefalos never forgave himself over the death of Procris, and he committed suicide by leaping from Cape Leucas