Warning

Well, you have a disbelieving spirit in your breast, if by swearing I cannot convince you or persuade you. But come, let us make an agreement, and the gods who hold Olympos may be our witnesses in time to come. If your king returns to his house, you will give me a cloak and a shirt as clothing and send me to Doulichion, where I would like to go. And if your king does not come as I predict, set your slaves on me and throw me from a great cliff, a warning to any other beggar about speaking deceptively!

Strabo lived 2000 years ago and called it an ancestral custom, so it could have been a costum even in Odysseus days. Menander lived 2400 years ago and wrote about Sappho taking the leap, but he was a play writer and might have invented the story.

Luckily Homer writes about this costum also and that probably is the reason why both Strabo and Menander did write about it.

This is what the 'stranger' Odyseus tells Eumaios when he refuses to believe that Odyseus is to be coming home soon:

Odyssey Book 14 line 382-399 Barry B Powell

The cliff

So yes, it seems to be a costum in Odysseus' days already. The custom fits Homer's writings and Odysseus' times perfectly. Strabo was right and Menander used the costum in his plays.

A good reason to accept this costum and see it fits the story.