The military does not have an alternative that is eliminating its need like the post office.
Mail is slowly going away and eventually, the right thing to do is have the gov't subcontract the service to FedEx (which it is already trending towards).
There is no need for a large bureaucracy anymore to support a function fewer and fewer people use.
FedEx provides air transportation services for the USPS. UPS also does some of this. Neither handle any delivery of mail or packages. In fact, both FedEx and UPS utilize the USPS delivery network to deliver some packages via their SmartPost and Basic service options, respectively.
With the rise of Blackwater type security firms, who knows? In fifty years we may be reading complaints about having to pay for benefits and pensions for "lazy gubmint soldiers."