It is always useful to make packets out of newspaper to hold loose fruit or flowers which would otherwise easily fall out and be lost. As you are breaking down the press, hold the open side of the newspaper higher. Then loose pieces will tend to slide to the safer folded edge. If there are many loose flowers or fruits, tap the edge slightly to move the parts closer to the folded edge. As you are breaking down the press, open the first sheet of each collection and put the remaining sheets in as you are doing it.
Much more time is saved later than is lost doing it.
Each sheet is most easily moved in the direction of the fold. That way the newspaper will not flop open because of movement through the air. Once the plants are all dry, put them into numerical order. Then all specimens of a collection will be together. It will save time and confusion.