My initial thoughts are that perhaps the ebay or your measurement is off. But we don't have the pleasure of having them all side by side to determine that.
So if we accept both measurements as being correct, we have a puzzle, don't we?
What would the purpose of your hat be? The design of your hat is a match (sorry about the pun :lol:) for the condiment set which was likely produced at somepoint after 1950. This time frame would rule out match striker and at 1.5 inches toothpick holder is out as well. An open salt is also very unlikely.