Convinced that the pictured wrench was not a truck wrench, I sent a picture of it to the webmaster of the White Steamcar site. Below is his answer:
Bob,
Thank you for your query. That is indeed a White Steam CAR wrench. It is for a White 1909 Model M car. That wrench was used for both removing the hub caps and opening the gas cap. Since the gas tank was pressurized to about 50 pounds this is a very important wrench to have. All White parts, including the wrenches, have part numbers. They all end in the model number, in this case a Model M. The Model M's were made in 1909. I have a Model M and would love to have this wrench if you ever consider parting with it.
Thank you
Mitch Gross