I would take a two pronged approach. In addition to your particular known song, even if it is in another genre that is outside your comfort zone but still speaks to you. Or studying the music of the publicly known masters of a genre there is a second prong.Is there a particular song or album that you've always loved, but is a little outside your typical comfort zone? If so, that's a good place to start!
Or, jam with some musicians outside your usual genre. Are you a rocker? Try going to an acoustic open mic. Or do you typically play reggae? Try sitting in at a blues jam. Etc.
I would take a song that I am already familiar with and listen to how genre specific cover bands handle it. It might not be a purist stab at the genre or the greatest of the genre that deep dive from other post describe, since it is a probably a local band adding their video to YouTube that is serving as my style guide. But it does allow you to start to relate the feel of the genre to something you already know well and then you can take that feel to a more pure take on other genres.