You don't need a monopoly. But if the government/quasi-government entity is effectively letter delivery of last resort while private companies pick off the more lucrative routes, they're going to either need subsidies or price delivery at a level that it may be untenable.

ADDED: IOW, you could have an external taxpayer-supported "rural mail delivery" department that has a charter to deliver a letter for a quarter anywhere in the country rather than providing a subsidy within the system.

It's mostly academic at this point. (And the reality is most private package delivery is fairly universal because there are business advantages to having a fairly complete delivery network even if it's not necessarily the most efficient decision at the micro level.)