How does one make olive oil that captures the taste of the olive?

I live in a Mediterranean country, where olive trees grow in public parks and boulevards.

Sad to say, the best olive oil in the country is exported for hard currency.
Is it possible to make superb olive oil from ‘standard’ olives?

If you DON’T know, please DON’T answer.
I ALWAYS get cold-pressed oil, with acid content < 1%. That doesn't guarantee taste.