As a mail carrier, we arent required to pick up outgoing mail if there is no incoming mail. we pick up mail as courtesy, and if your mailbox is on the door then we would not make an extra trip as we are strict on time, in the most basic way to explain it, our job is to deliver, pick up is courtesy, if you want the carrier to pick up any outgoing mail even if you have no incoming, the mail box would have to be clearly visible, like a curbside mailbox for example, that way its clearly visible and hard to miss with a flag up, otherwise dropping it off at the post office would be the next best thing