There are two important things not yet mentioned here. The fife must be capable of high b. I learned fife on a Ferrary. It was a wonderful instrument and played a high b easily, as long as the instrument was properly tuned. Many years ago I posted (ok, so it’s been a while) some information about how cork placement and embouchure affected the tone. This also has an incredible impact on the ability to hit the higher register notes.
When I purchased my Model F, I pulled it out of the packaging and played all the way up to the fourth d’. Though I loved the Ferrary, this was only achieved with a force of will. The Model F was just simply built to play the highest register notes (and all the others as well - can you tell I love the instrument?).
I also have some inexpensive Cooperman maple fifes. They play very nicely in the lower registers, however, the highest notes are hard to hit. The plastic Coopermans play nicely in the higher register.
I have a keyed fife made by a flute player that just does not have clear high register notes (this was a second he sent as a favor, so I will not mention the brand. He does make wonderful flutes.)
I have found that cork placement makes a big difference. My Model F, Cooperman and Ferrary all played nicely with the cork just shy of the hole. It takes some tweaking, but it is worth it.
Rather than just blowing harder, don’t block the hole with your lip and learn to blow sharper. I do not know a better way to describe it. I pull back the corners of my mouth and adjust to avoid widening my lips. It takes less breath and produces a loud clean tone.
I thought of a third thing as well. If the fife is wooden, soak it in oil. I have seen many people recommend different kinds of vegetable and seed oils, but I have always used mineral oil. It won’t rot, won’t smell and you can find a large bottle at the pharmacy that will last forever. Just wipe off all of the excess oil. This made the difference when I was first learning fife. The stock fife I used had not been oiled in years and was very dry. Oiling it gave me my first clear high b’ experience.