What is the simplest way to approximate time after the sun has set? Perhaps using the moon's height and phase? This would make the most sense to me, but what could you do in the case of a new moon?
Another way to find the time is to use well known stars. In the northern hemisphere, you can use the Pole Star and the Big Dipper to tell the time fairly accurately. A good explanation of the procedure can be found here. Here is a abbreviated quote from that site :
The website also gives an example, so you can work trough it too make sure you understand all the steps. You do need to remember of few things, but once you get the hang of it should be pretty straightforward.
I looked online and apparently for the southern hemisphere there is a similar method using the Southern Cross.
If you know the stars, look at the constellation in the Zodiac. For example, according to astrology, now the sun is in Aquarius. In reality, it is one constellation backwards, so it is in Capricorn. Then, look at the starred sky - which constellations of the Zodiac can you still see? E.g. if you see (left to right), Leo, Cancer, Gemini, Taurus and horizon, then you know that there is still Aries, Pisces, Aquarius below and the sun is now "at the end" of Capricorn. So actually if you count cca 24/12 = 2 hours per Zodiac sign, then it is cca 6 hours after sunset. This is simple, logical, doesn't require moon, nor any "calibration".
But if you look at the starred sky every night, it is much easier not only to orientate yourself, but also to guess time. E.g. in winter the Orion dominates the evening sky. So if you see it leaning steeply to his right side and low above horizon, you plus-minus know where the west lies, and also it indicates a coming midnight. And if you see the Lion leaning his head just above horizon, you know that you really spend too much time in the pub - the new day will start soon. :) Of course this more or less is valid now, in winter - the situation changes over year, so you must regularly calibrate your star clock. :-)
An astronomer with a protractor and pocket-calculator can figure almost anything. Unfortunately, though, astronomers don't fit very well in survival kits (and grumble when you try to stuff them in there). So normal people are handicapped, for sure, but we can still say some basic things.
A full moon is opposite the sun, so if you know it's a full moon tonight, and if the full moon is directly south, then you know it's right around solar midnight, that is, 12 hours past solar noon. It's not exact, though, because the moon doesn't trace through the sky at exactly the same speed as the sun.
If you're familiar with the area and practiced, you could use the full moon to estimate hours other than midnight and not be far off, say, within an hour. Note, however, that a sundial (moondial?) trick wouldn't work without some fiddling because the moon is only rarely in the same plane as the sun. The gnomon would need to be adjusted to the right angle - something an astronomer could figure.
Similarly, if a First-Quarter moon is directly south, then you know it's right around 6 hours past solar noon. If a Full moon is south then it's 12 hours past noon, and if a Third-Quarter moon is south then it's 18 hours past noon, that is, 6 hours before noon. A New moon that's south is zero hours past noon, so you can't see it, and it's useless for telling time at night.
None of the above is very precise, but it can give you a rough guide to use when you wake up in the middle of the night for a bathroom break and forgot your watch back in the sleeping bag.