Now .. I am no authority on teaching ...
Here is how I was taught to do it by reading a book. After placing the piece on your work and applique sewing it down by hand, with almost an inch or two left in the opening of your work, put your needle into the hole and to the top... Then into the paper to catch the paper only, pull down and it will release it... keep pulling down and it usually will completely drop down inside the applique. Then with your fingers or a tweezer, pull it out and complete the seam.
I will say here .. the trick is not catching the paper while sewing it down. The tiny stitches must be catching only a couple threads on the outside of the piece. Although, I have managed to release it when my stitches pierced the edge somewhat.
Hope this helps ...
All that being said, you can use a piece of interfacing and sew the piece right sides together with tiny stitches and tiny seam allowance, slit the interfacing and turn the work clip the corners or any sharp curves being careful not to clip into the seam and go from there. The interfacing leaves very little weight and no bulk to speak of.