That is a luggage rack. It was very popular on the model T, which had no driver's door, so it did not interfere with entry. Some older cars had doors that were so high from the running board that they would open with the rack in place. It is not very useful on a model A, as the doors will not open with it in place.
There are pictures of those racks with lambs,goats,pigs,and dogs tied in them.I came across them pawing through old collections of Life magazines,looking at articles on the depression.I bet somebody clever with the internet could scout some of those pictures up.