Here is what I get when I normalize the whole file :
You can boost something like that just fine. Look for the "Normalize" filter/effect/button in your audio program.
As you can see, there's still a major difference in volume between the two parts I previously encircled in red and the rest of the waveform. The thing is, I'm trying to avoid having to adjust the volume of the rest of the track manually (as Data suggested). Is that possible ?