Hi there
first of all, I should say I was having a problem to make a higher notes on my recorder " C Yamaha, Soprano" but I searched on internet and here are some tips that I think it would help for producing the higher notes, (I don't know the author):

If you are having trouble playing high notes consider this:

Make sure that the high E on the C recorder, is played with the thumb creating a thin opening. This will set you up to play all the other high notes above this better than if you use no thumb at all.

Start by closing the thumbhole completely. Then pull down and lightly press in with the upper left corner of the thumb. Use the side of the thumb just before the edge of the nail. Keep the opening thin.
Try to be conscious of this small opening and keeping it that way as you go up the scale from the high A or E respectively. As you make this small movement your knuckle remains relatively flat. If you flex your knuckle you will feel tension and end up using too much of the thumbnail its self.

Use of a thumbnail technique will eventually damage wood recorders.
Using this method, you will be able to play all the way up from A to high G (on the f recorder) or E to high D (on the C recorder) without any change in the thumb position.