This is, perhaps, the opposite. Touring the World (as I do) by Google Maps, I was delighted to discover the town of Accident, Maryland. There is a grid plan in what must be the original settlement. The main drag, running roughly north to south, is called Main St. It's extension to the north is called North Main St and that to the south South Main St.. The parallel road to the west is in two parts called South North St and North North St. North North St turns east at the top to join Main St.
South St branches off Main St near the centre of town heading east until it reaches the eastern side of the grid plan which is divided into North South St and South St. There is no South South St because South Main St takes a jink to the left to join the southerly end of South St. The road that continues the line of South Main St heading south is called Main St Exd. I have no idea what that means.