I recommend being very specific about the order in which they are passed.
If you are worried about grade privacy just put the grade on the back of the paper instead of the front. The end of class is typically hectic; you are rushing to finish before the bell because everything took longer than expected as it will nearly every day your first few years.
In middle and high school passing papers back as efficiently as possible starts with how students pass them up.
The way you act, where I come from you'd never last Could it be you're lying through your teeth? For example, if students are in rows the process starts at the very back, then each student in the row should put their paper on top and then pass it to the person in front of them.
You want a student doing the job instead of doing it yourself because once you give a Type-A student a task, she will be laser focused and that bin will be emptied everyday like clockwork. The process ends when there is one stack on the desk of the student in the front row on the right side of the room.