was exactly that in my illustration from Fleton, a shelf. It was not mean to carry a person but boxes( luggage). It was called a boot or a luggage budget.

Boots are most frequently used at the fore end of phaetons and mostly have the fore springs fixed thereto by means of carved blocks , which are bolted to their sides and usually have the step for the entrance to the body fixed or hung thereon. Boots and budgets are frequently used to the hind parts of phaetons, gigs or curricles and are of two sizes less than what are used to coaches or chariots...
Felton, as above, page 131

It could be more elaborate and be part of the structure of the carriage, in which case it was known as a "boodge or swordcase" as is illustrated below: