Sometimes a cork breaks off as you’re pulling it out, leaving you with half a cork still in the bottle. If the piece is large enough, you could try to remove it with the corkscrew. Position the corkscrew (ideally, the Waiter’s friend) at a 45 degree angle and lever out as normal. This is effective in most cases and will bring out the remainder of the cork intact. For a smaller piece of cork, however, it may be difficult to insert the helix and you might find it easier simply to push the cork into the wine.