Steve Leggett, in his review of the album for AllMusic, called "Cover Me Up" a "beautiful opening love song". Writing for Paste, Jerrick Adams stated in a review of Southeastern that " on the one hand a gentle, insistent love song, and on the other a moving testament to personal redemption that never once turns a blind eye to past indiscretions. It sets the tone for the remainder of the album, which is given equally to the promise of romance and the ever-looming possibility of suffering, both self-induced and arbitrary." In 2020, Paste ranked the song number one on their list of the 20 greatest Jason Isbell songs, and in 2022, American Songwriter ranked the song number four on their list of the 10 greatest Jason Isbell songs.
