some strategies I use to be able to play tunes in the written key, that have notes below the root, usually a 4th.
if the tune is in sharps, I play the next Key up, so if G, I play with a D flute and flat the C# (half hole the top note), and I play the root note on 3.
if the tune is in flats, I use the flute below the written key, so if Bb, I use F and flat the E to get my 2 flats, and I play the root note on 3.