Hmmmm. I haven’t worn the E-6 and below uniform since 1978, but If i remember correctly, the undress white uniform had no piping and the service dress whites included piping, both required the neckerchief.
I was an E-5 in 1965 and undress whites was white pants and a white jumper. A neckerchief made it dress whites. There was no piping. That was only on our dress blues.