The Moon is the good news placement. It spends about two and a half days in each sign, so unlike the rising sign it is not destroyed by a missing birth time. The catch is that on roughly 12 days out of 30 the Moon crosses a sign boundary, and those are exactly the birthdays where the question gets hard.
The Moon travels roughly 13 degrees per day against the zodiac, completing a full circuit in about 27.3 days. Divide that by 30 degrees per sign and you get about 2.3 days per sign on average. Its actual speed varies from about 11.8 degrees per day at apogee to 15 degrees per day at perigee, so a slow Moon can linger past two and a half days and a fast one can clear a sign in under two.
The practical consequence is simple. If the Moon was solidly inside one sign for the entire 24 hours of your birth date, then your Moon sign is known, full stop, with no birth time required. Only the exact degree is unknown, and the degree matters far less than the sign for most interpretive purposes.
You need an ephemeris or any calculator that takes a date without a time. Set the location to something arbitrary, because the Moon's sign on a given date is the same worldwide with one exception noted below. Then read the Moon's position at 00:00 and at 23:59 of your birth date.
If both ends of the day show the same sign, you are done. If they differ, the Moon changed sign during those 24 hours and you are in the ambiguous group. There is a third case people miss: the Moon can be at the very end of a sign at the start of the day and cross into the next one within hours, which narrows the ambiguity to a much shorter window than a full day.
When the Moon changed sign on your birthday, you are choosing between two descriptions that can feel very different. A Cancer Moon and a Leo Moon both want to be seen, but one withdraws to be soothed and the other performs to be praised. There is no reliable shortcut that turns this into a calculation.
What you can do is read both and notice which one lands without effort. People who have lived with a Moon sign for decades usually recognize it fast, because the Moon governs instinct, comfort, and emotional reflex, and those are things you have direct evidence of. This is soft evidence, and it should be presented as such, but it is not nothing. Avoid treating emotional self-recognition as proof, since both readings will contain some statements that sound true.
The Moon's house placement is gone without a birth time, because houses require the Ascendant. So is any aspect from the Moon to the Ascendant or Midheaven. If the Moon changed sign that day, the Moon's aspects to the faster planets may shift too, since the Moon moves 12 to 15 degrees across a single day and can move in or out of orb with Mercury or Venus.
What survives is the Moon's sign, its aspects to the slow planets, and its phase. And the phase is underrated. New Moon, First Quarter, Full, Last Quarter births have a documented feel in the tradition, and the phase is determined by the Sun-Moon angle, which you can approximate from the date alone.
If the Moon held one sign all day, treat its sign reading as you would in a fully timed chart. It is the placement that governs instinct and emotional reflex, and it does not become weaker just because the clock is missing. The loss is confined to arena, meaning where in life that reflex plays out, which is house territory and therefore out of reach.
Suppose the birth date is the 14th and the ephemeris shows the Moon at 27 degrees Taurus at midnight and 9 degrees Gemini at 23:59. The Moon crossed into Gemini during that day. If the only memory of the birth is that it happened in the morning, Taurus is the better bet, since the Moon was still in Taurus for the first part of the day. If the memory is evening, Gemini wins.
Now suppose the ephemeris shows 2 degrees Cancer at midnight and 15 degrees Cancer at 23:59. The Moon was in Cancer the entire day, so Cancer is your Moon sign with no time needed. Those two cases cover most birthdays, and sorting them takes one lookup rather than a chart calculation.