The Moon
The moon is advocated by some references as being suitable for improvised navigation. It is, but only in a similar way to the sun. It is a better "sense of direction" indicator than a reliably accurate direction indicator.
The methods generally described for navigation by the moon are the directions of rise and set; and direction indicated by the phase of the moon. The shadow method used to navigate by the sun can also be adapted.
The moon's orbit is tilted about 5 degrees from the ecliptic (the orbit of the earth around the sun, or more correctly the apparent path of the sun across the sky). This means the rise and set directions for the moon can be up to 28.5 degrees either side of true East/West (compared with 23.5 degrees for the sun). Of course, the error due to latitude is not included in that! (Because of precession, the angle of the moon's orbit to the earth's equator actually changes from about 18.5 degrees to 28.5 degrees over a period of several years).
So, you see the moon is actually a pretty inaccurate indicator of direction.
The one improvised method, that is accurate for navigation by the moon is the shadow stick method. However, as the moon is at its zenith (highest point in the sky) at different times from day to day, it is important to plot the points when the shadow is the same length each side of the zenith. A line from the first point to the second point indicates East. This method is most practical on clear moon lit nights in order to get a distinguishable shadow. The stick used to generate the shadow must be vertical for this method to be valid.
Normally the moon is most useful as a rough "sense of direction" indicator. It rises nominally in the East, and sets roughly in the West. If the moon is following the sun across the sky (ie visible early evening) the illuminated side of the moon is very roughly West. Likewise, if the moon leads the sun across the sky (ie doesn't rise till after sunset), then the illuminated side is roughly East. At least one reference I have seen suggests this navigation by the phase of the moon is accurate, but it is not. It is subject to errors due to the seasons, the inclination of the moon's orbit, and the observer's latitude.