The mail delivery system will be sold and/or leased to private carriers such as UPS or Federal Express.
For rural areas that UPS cannot afford to profitably service, post offices can contracted to local businesses, such as convenience stores. UPS would sort the mail and deliver it to the local post offices to be placed in the individual boxes. This would draw traffic into the store and give the business additional income. This is how rural post offices were handled up until the mid-1970s. This was a much cheaper method of handling rural mail delivery than building post offices in every small community.
For rural mail delivery to homes, UPS could employ contract carriers the same way the US Post Office does now. This will employ local citizens which will help the local economies.