How about "Poor Sweet Baby" from Snoopy!!!, or "I can't Say No" from Oklahoma!. Well I guess it depends on whether the director wants Little Red to be cute or sexually awakened by her experience with the wolf. I've seen it played both ways, and think the later works best.
If I were directing "Into the Woods" (and I guess this would go for any show), I would want someone to bring something new to the table...portraying Little Red as "Cute" isn't new or interesting, and may leave your director cold...I would encourage you to find some kind of material that shows off your talent and your ability to bring something fresh to the role...
The lyrics of "I know things now" seem to equate Little Red's experience in the Wolf's belly with the loss of one's virginity...Notice how she carries a knife after the experience and looses trust of anyone around her...I've seen it played that way and it works great...I though Danielle Ferland played it that way, even if she didn't really know that was how she was playing it.
I've played Little Red three times... once in my high school production and twice professionally. When I auditioned at my high school, I was 17 and sang "Shy" from Once Upon a Mattress. When I auditioned professionally, I was 19 both times and sang "I Know Things Now" at the first audition (at the request of the director) and "Good Morning Baltimore" from Hairspray at the second one.
I think anything belty that shows acting talent will help get you the role. "My New Philosophy" is cute but there is SO much more to Little Red than being cute. She is very obnoxious, which is why everything she sings is belted in that high obnoxious kid belting range. I once saw the show done and the girl who played the role had a GORGEOUS voice - but it totally didn't fit the part. "I Know Things Now" sounded like an aria, lol, and the whole point of her music being in that range is to complement the character's personality (also why Cinderella, the more ingenue, has a light soprano sound, and why the Princes have the high tenors, etc).
Little Red is more complex than most people think. A lot of people have the misconception that she is supposed to be an adorable little 12-year-old, but she's not. She's a charicature, but you have to play her as a real person, because everyone in this show are real people with real emotions, even though they seem so over the top. So I would just pick a song that shows a nice belty range that you can have fun with and make real acting choices with.
I'm auditioning for little red too and a lot of people have suggested my new philosophy to me as well. This is kind of a dumb question, but when you sing a duet like my new philosophy at an audition, do you sing both parts or just the main part? For example, if you're female and just wanting to sing sallys part, would you just let the music go for schroeders part?