All that having the wall lit by the sun at 12 noon does is to allow you to identify the direction of due South.
I assume that you are using the technique where at the instant of 12 o'clock local-apparent-noon you mark the line of a shadow of a vertical plumb line on a horizontal surface adjacent to the wall (like on the top of a table pushed against the wall) and that you then measure the angle between this line and a perpendicular to the wall in order to find the wall's declination. If you need to establish the declination of a wall that is not lit at 12 o'clock local apparent noon then you need to make the measurement at a time when the sun is exactly a known angle from due South,say South-West. This then gives you a line from which you can construct the meridian and the declination can be found as before.
