Yup, psychoacoustically, at lower volumes you don't perceive the bass and highs to be at the same volume as the mids. But, when you raise the volume, you start hearing relatively more bass and highs in relation to the mids, and at about 100db, things start to sound roughly flat in terms of frequency response.
Not that you should listen at 100db, mind you. But, it's no surprise that, when you raise the volume, the music sounds fuller.
There's nothing wrong with liking what you like, and you don't need to apologize for it. I would say that any Bose headphones I've heard thus far haven't been particularly revealing, and they certainly aren't difficult to drive, so I would suggest that spending more on a portable amp - especially an inexpensive amp like a fiio - would do very little for you. A source upgrade might help, but again, unless you have very resolving headphones you probably won't appreciate it enough in order for it to be worth the money.
Now, if you want a headphone upgrade, then that's a very different story. Although not all headphones are created equal in terms of resolution, there certainly are those that will very quickly show you how inadequate the vast majority of sources are. I would say that if you want a headphone that's smooth, warm, reasonably bassy, but quite revealing at the same time, the Sennheiser HD600 or HD650 is in your future, but those are headphones for which you will certainly need a desktop amp as well as a quality source, and there, any upgrade in your signal path will be readily audible.
In the meanwhile, upgrade your music. Going from mp3 to flac with a pair of Bose may not pay off huge, but seeking out better-recorded music certainly will. Get a dynamic range meter plugin for Foobar or whatever player you're using, and see for yourself how much less dynamic range those modern remasters have than the originals from the 80s and 90s. Taking the extra steps and seeking out the best recorded version of something is usually an audio upgrade that's immediately rewarding, not to mention quite cheap.