I don’t really care for “Rock-a-bye Baby” or “Twinkle Twinkle Little Star.” I’ve been singing/humming “Ring of Fire” by Johnny Cash and “Poem on the Underground Wall” by Simon and Garfunkel.
What songs are you singing? I’d love to add more to my rotation for the baby!
We sing - I Wanna Hold Your Hand and Obladi Oblada and Hey Jude.
What an interesting question, a non traditional song that you can sing to your baby is, “What a Wonderful World” by Louis Armstrong, 
These song is always soothing and meaningful and, often carrying a personal significance for the parent, which can add a special touch to the bedtime routine.