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.
There is a common and intuitive response to reaching for the third octave. When it does not come easily, players will effort more: tensioning the lip muscles more and blowing harder. This makes sense intuitively but it doesn't work as a strategy. The third octave requires no more lip tension or breath pressure than any other octave. The embouchure needs a slight flattening of the blowing aperture. This will do the job.
A good way to practice is with octave jumps. Start with a well-made lower octave tone and by using the LEAST amount of effort and air, practice jumping up to the next octave, slighly flattening the embouchure, but applying no extra push. Play quietly, then you will find the easy relationship between the octaves.
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.
Lion's Breath
One way of deliberately breaking the muscle tension of your embouchure between phrases, is to "lion's Breath", familiar to yoga practioners. Open the mouth and eyes as wide as possible, relax everything in the lip, jaws, throat and stick the tongue out. Then, re-make your embouchure just before intoning the next note. This is probably best used just for practice, not performance!