Most of us use the FG3 trill that we first learned where we finger F3 and trill with our thumb. That trill is useful in most situations, but has the disadvantage of yielding a flat G3. What if we needed to play that trill in the orchestra in tune with an oboe?
Many flute players have requested this fingering repeatedly in the FLUTELIST forum, which I posted a few years ago (2004-09-22).
Here is a fingering that you might want to try:
Note: rather than pressing down the key-ring and opening the entire half-hole key in the left hand, experiment with covering about one half of the open hole, or cover the hole completely and see if you prefer the way it sounds.