You don't need a worker. See this example in the rules:
The first player chooses the action “buy hull parts”. He receives a bonus of 2 Victory Points and pushes the Action Tile down to cover the bonus space. Then he performs the action “buy hull parts”. He buys all 4 hull parts on the Action Tile, and therefore has to pay 6 Taler and employ 4 Workers. 2 blue Workers are available to him from the Turntable, so he still needs 2 Workers from his personal stock, so he puts them back in the
general supply. Because he bought all 4 parts on the Action Tile, he gets a
free hull part of his choice, which he puts into his Warehouse.
Note that there is no mention of a requirement for a fifth worker for the free hull part.