Massage with vegetable oil, usually mixed with some fragrances, had been a widespread practice for many millennia, not only in Ancient Greece, but in most lands around the Mediterranean, which was about as frequent as it would be today to take a shower.
In fact, in the ancient world the main use for vegetable oils was for massage and for perfumes, and not as food.
In the ancient literary sources, there are very few, if any, mentions of vegetable oil used as food, but countless mentions of massage with oil.
Already in the first version of the Gilgamesh Epic, almost 4 millennia ago (the Old Babylonian version), there were 4 pleasures listed as the benefits of being a civilized man as opposed to a savage: making love with a professional woman, eating bread, drinking beer and being massaged with oil (these were used to lure Enkidu into going to a city).
While in later times olive oil was the main oil used for massage, in the Gilgamesh Epic it seems that the oil that was used was sesame oil.
2000 years after Gilgamesh, e.g. in Pliny the Elder, similar accounts were given, i.e. that the main benefit from grapes is drinking wine while the main benefit from olives is being massaged with olive oil, both for pleasure and for a healthy skin.
While massage with olive oil or other vegetable oils was ubiquitous and daily for those who could afford it, for me it is a bit of a mystery how they cleaned themselves after that, in the absence of soap, because I have never seen any mention about this.