Relaxed third octave playing
Perhaps counter-intuitively, the third octave can be played with a relaxed embouchure. Third octave notes are no more difficult than any other note on the flute. Don't blow hard...concentrate on bringing the top lip a little closer down to the bottom lip, thus flattening the embouchure opening. Usually, this is sufficient to kick the sound up an octave, without blowing a gale into the flute. It's all about embouchure shaping.
Surprisingly, the third octave notes are easy to play as very long tones....they need little breath.
Fingering inconsistencies amongst instruments tend to show up more here than anywhere else, and mostly in the upper half of the octave. There are also variations in the note names too. San go no re is probably at the upper end of anything in the classical repertoire. Higher notes might be used for some sound effects, and modern compositions occasionally explore the highest range. Some flutes don't perform very well in the upper half octave. This may depend on the style of the flute and the particular emphasis of the maker, and is not necessarily a problem.
One way of deliberately breaking the muscle tension of your embouchure between phrases, is to make a ‘fish-mouth’ shape with your lips. Open the mouth, relax everything in the lip, jaws, throat and then re-make your embouchure just before intoning the next note. This is probably best used just for practice, not performance!