Name of extra skin
The skin that continues upward and remains connected to you inner labia is just that, an extension of you inner labia. Regardless of name, it too can be trimmed along with the lower larger parts during labiaplasty to create a more homogeneous result. Glad to help.
The skin you are referring to is an extra fold of your clitoral hood connecting to your labium. I am the inventor of the central wedge labia reduction technique with the hockey stick clitoral hood extension. You are an excellent candidate for this procedure which will make you look normal if done by a skilled surgeon. The surgery takes about one and a half to two hours.
The extra tissue is called a labia minora reduplication. This is extra labia minora tissue that is common, particularly when enlargement of the labia is present. The best procedure for this would be an extended wedge technique described by Dr. Gary Alter. This would allow reduction of the labia minora and removal of the extra tissue, sometimes called clitoral hooding. If you are OK with the appearance and size of the labia, it is certainly possible to remove only the reduplication. Make sure that YOUR goals line up with your SURGEON'S goals. One size does not fit all with vulvovaginal surgery, and the most unhappy patients are the ones who's goals weren't lined up properly with the surgeon's abilities and techniques.
Labia Minora Reduplicaton at the Clitoral Hood
Your photos demonstrate a common outgrowth (reduplication) of the labia minora along the sides of the clitoral hood. The cosmetic fix for this appearance is commonly called clitoral hood contouring or reduction even though it's not really the clitoral hood itself that is the problem. The tissue is removed with linear excisions along both sides and sometimes over the top. It is sometimes combined with labia minora reduction, but it can be done alone.