The problem is that eventually, the cork starts absorbing the oil and gets heavy and heavier, eventually sinking into the oil and smothering the wick.
Cork has a SG of about .16 and olive oil has a SG of about .91. No matter how much oil is absorbed by the cork it will never get to a SG of higher than .91.
The solution- leave the old wick in! the tip of it will push the new wick up long enough to burn a significant time.
I have no idea what that will accomplish. (Maybe the problem is that I’m not even sure what you are suggesting.)