Do you have an idea of what you want to upgrade/change other than trying open headphones? More bass? Louder? Etc.?
The main difference with different headphones is frequency response, and to a large extent that can be adjusted with EQ. (Your interface will have flat frequency response... unless "loading" it with low-impedance headphones messes-up the frequency response.)
Of course you can monitor/listen with your setup, but audio interfaces (and microphones) are more for audio production than for reproduction.
And with reproduction (and monitoring), we generally just want accuracy to accurately reproduce drums. bass, ambient, classical, heavy metal, or whatever is playing.
If there's a difference with different styles of music, you may want more power/loudness with certain kinds of music. Some genres have very little bass, but usually we want "full range" capability, no matter what we are listening to.