> boost the clarity of softer sections or to make louder sections more comfortable to the ears
You can create these effects during playback with a compressor, which are available in many players[1]. Knowledge about compressors isn't as widely known as it should be; it's an important feature that would help a lot of people, especially anyone dealing with hearing loss.
> headphones
Maybe try better headphones (or a different type, such as enclosed headphones with good room attenuation)? Many cheap[2] headphones have terrible response to different volumes; quieter parts may be overly attenuated, for example. Cheap/bad headphones also tend to have a highly variable frequency response, which can sometimes sound like volume problems. If this is the case, spending some time with a highly configurable EQ might help (possibly in addition to a compressor).
[1] also available in some audio systems (e.g. jack) and as a feature in some drivers/soundcards
[2] and some "name brand", such as older Bose, anything Beats or other brands sold as fashion statements.
edit: re-added sentence that died during a bad copy/paste
edit2: Appendix for anybody that isn't familiar with dynamic range compression.
> need to adjust the volume several times
This is literally what a basic compressor is automating[3] for you. It turns down the output volume when the input level is over a threshold. There are a lot of (often configurable) details like how much to reduce the volume as the input level increases past the threshold, how quickly[4] it responds to a loud transient, and how quickly it returns to normal when the input becomes quiet again. Really nice compressors even smooth[5] the changes near the threshold so the whole effect is less noticeable.
[3] https://en.wikipedia.org/wiki/Dynamic_range_compression#Desi...
[4] https://en.wikipedia.org/wiki/Dynamic_range_compression#Atta...
[5] https://en.wikipedia.org/wiki/Dynamic_range_compression#Soft...