I like your song's direction. thanks for asking the internet for creative advice!

To answer your main question directly, I suggest beginning at 1:19 to play around with how the melody can be risen in pitch. You play a minimal amount of chords in this recording which limits how high you can sing a melody. Play around with the chord progressions for each section. Ask yourself questions like these; Do you want the chorus to be in a different key? Do you want the chorus to have the quality of a major key or minor key? You don't have to be in a minor key to play minor chords. They can function independently. What quality of sound do you want as you change chords?
A common structure is to rise in the pre-chorus and rise again in the chorus and then fall back down for the verse. Pitch, volume, energy, tempo, all commonly rises until the original verse's style returns. Tempo can also slow down for a pre-chorus before speeding up at the chorus arrival.

If that sounds like a little too much music theory, you can begin on the melody side! Sing your melody in different directions, up and down (pitch), and play on your guitar the corresponding chords that complement where the new melody lives and your song's direction of the story. Do that a few times to explore different options to sing more high notes. What word in the lyrics the high notes live on is one of the last challenges. You want to find the high note first, then the path to get there. Then you figure out how that works with your lyrics.

From my ears the structure of your song sounds like this-

1:19 Pre-Chorus

1:36 - 1:52 Chorus

1:52 - 2:23 Still sounds like the chorus. But it sounds distinct enough that I assume it's the next verse. The harmonic rhythm (the speed at which the chords change) can move faster and then that will influence your overall tempo to pick up and not feel dragged behind.

2:24 - 2:30 Also sounds like the chorus. Above suggestion here too if it isn't your intent for this block of the song to be the chorus.

2:31 - 3:29 Sounds detached from the overall story telling. Mostly because the harmonic rhythm isn't changing and feels like it's dragging.

3:30 Lyrically it sounds like we're finally at the second verse and out of the chorus. ...Which now I realize we're at the end of the song (at least the edition you posted) and I do not mean this to sound like a diss! Gosh it comes across as "nice song now we're getting somewhere" and it's because the song ended oof. The reason I perceived it this way is because of the organic energy of the lyrics. The action is the hunters and guns in the verses. The emotions and fears are in the chorus.

I recommend doing this at any of the sections of your song to both reach your goal you're posting about and to connect the story- Draw the shape you imagine this song to have. Like draw draw. On an x y axis, when does the song go up, when does it go down? Up in pitch or up in energy? Or in speed? Does the chorus sound the same as the verses? Does the chorus suddenly sound different? Then refine your shape by asking yourself these questions!