Obviously, having a post office was vital to new settlers coming into an area, so typically as soon as there were about a dozen households in an area that was inconveniently distant from an existing post office, they'd apply to the Post Office Department (it didn't become the Postal Service until 1970) for establishment of a new one. Because people coming to pick up their mail were potential customers, a grocer or tavern keeper would generally be delighted to perform the necessary duties, and would make the application with himself as potential postmaster of the new office.
The application included information about the exact location of the new office, the number of people to be served, its relation to existing rail lines and navigable waterways as well as its location on or near an existing mail route, the nearest existing post offices, etc. In general, a new office needed to be two miles or more from an existing one, but there were exceptions. No population minimum was specified, and the Post Office Department tended to be liberal in establishing new offices.
A name would also be proposed for the new office, which might be rejected if too similar to an existing office in the same state, and that led to all sorts of placename curiosities. One of my favorites is Elburn, Illinois, which first applied to be Melbourne—but was obliged to lop off some letters when an existing Illinois office turned out to already have that name. The local newspaper's masthead still notes that it's "the only Elburn in the world."
The process for establishing a new post office is explained in this 1893 book and, in fact, remains much the same today.
Follow up: What are some books or articles about the process of founding a new town, not necessarily in the US West? Just the process of splitting from society to create and maintain a new settlement.
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