How do you get music on animal crossing not on a Saturday night?

You cannot. K.K. Slider will only appear on Saturday nights after 8PM in the Roost; however in New Leaf, he will appear in Club LOL by 8PM on weekdays and Sunday under the guise of DJ K.K to play remixes of his tracks, but you aren't able to request songs or get the tracks he's playing.

Alternatively, you can trade with other players.