Yes, you want to go to a size smaller needle. You want to fit more sts (16.5) into 4 inches than you're currently getting (16). See what the smaller needle gives you and if it's too many sts, you can adjust the number of cast on sts to give you the size vest you want.
You're close with the 5.5mm needles, only 5 extra sts over 40 inches which would be a little over an inch extra width. Don't worry about the row tension since the directions probably will have you knit for a certain length anyway.