I'm pretty sure postal-like system had already existed since classical age. During the Tang Dynasty of China, the postal system was even famous for the delivery of chinensis to please a emperor's queen.
In ancient time, communication system was much more costly than modern, as an result, stuff for public use such as governmental message usually had the priority to use it. It goes without saying that there must be. It went without saying
Bilbo, such a big guy could operate such public resources.
Remember this, most of all, in literature,
novels's key elements are those characters. As an result, LOTR series, typical novels, would mainly follow the characters's angle, those not his/her business or involving the plot's progress, wouldn't get mentioned in details. For instance,there'd be nothing but stuff relative with a soldier's rank and Tower-Guard's unit such as comradeship, detailed combat, high-command's security and so forth, when it came to
Pippin's angle in ROTK. Thus, just use some common sense then you'd mostly figure out how
Shire's postal system'd look like.
JRRT didn't have to insult the readers's intelligence and common sense, detailed settings wouldn't be mentioned too much thus, my dear

, unless someone dares smoke up people that him/her self as an literature's big guy, and even forcing other JRRT's fans to follow his/her settings.