I do considerable amounts of stack cutting. I make wind chimes as a hobby.
I use a paint brush to clean each piece, use painters tape or masking tape half inch and nothing wider, roll it together with my finger and put a few pieces on each piece.
I have a Black and Decker hobby vice that is plastic and rubber that will compress small pieces.
and, I also have been known to compress it by placing a piece of paper on it on the floor and stepping on the stack lightly.
Just for fun I will tell you what I do for the pattern. Seems nearly all glue is hard to remove from the top piece with the pattern so I use wide painters tape on the top piece. Then spray glue the pattern to the tape which I print out on my printer ....to the top board.
The tape in between I pay attention to my pattern so very small cut outs do not come in contact with the actual tape as when you pull it apart baltic birch may pull off the top layer of wood with it.
All the suggestions in this thread are good ones, this is just my way