"Run" expresses Swift's desire to go on clandestine escapades with her lover where no one will find them. It isn't the first time Swift has conveyed this sentiment. Several songs from throughout her catalogue, such as "I Know Places", "Wildest Dreams", and "The Lake", drive home how much she yearns for privacy while in the public eye. Although, in the context of "All Too Well (10 Minute Version)" — a fellow Red (Taylor's Version) track — "Run" may be about the Swift-Gyllenhaal relationship as well.
Lyrics from "Run" like, "Give me the keys, I'll bring the car back around" and "We shouldn't be in this town" seem to mirror the "All Too Well" lyrics, "You were tossing me the car keys" and "We were always skipping town." Altogether, they feel like lyrical puzzle pieces that create one complete picture.