There are several possible routes from tile 1 to tile 6. However, you can’t pass over the same tile twice, so you should probably follow the edge of the room before you make a turn.
The path from tile 6 to tile 12 will look the same as the path from tile 12 to tile 18.
The path from tile 18 to tile 24 is a straight line.
If you treat the vertical line that runs down the center of the room as a mirror, then you can clean each side using the same zigzag pattern that you already created!