NavList:

Peter Hakel, you wrote:
"How about using the Moon’s phase? "

Yes. It's a low-res lunar distance. The Moon's phase, quantified as fraction illuminated, is equal to the "haversine" of the lunar distance from the Sun. Or, ignoring the obsolete haversine function, the fraction illuminated is:
  f = [1 - cos(LD)]/2,
where LD is the angle from the Sun to the Moon. I talk about this in my "Lunars" class as an introduction to the concept of getting time by lunar distances. People have been judging the date by Moon phase for thousands of years. A skilled observer can distinguish one day from the next especially when the Moon is close to half full, but even casual observers can detect the change in phase over two days.

Of course there's a much easier way for a "modern" navigator. Assuming you have an ordinary Nautical Almanac, just plot the Moon's SHA and Dec on the little star chart in the back. You don't have to worry about parallax when your only goal is to get the date. The almanac lists GHA for the Moon every hour. You have to subtract GHA Aries for the same hour to find the SHA. Other than that it's trivial. 

Frank Reed