The Moon on this night of the story, now only a couple of nights from full, makes an appearance in 3 separate chapters across 3 books, and you can see a very similar moon tonight (2021 February 24)!
After their parley with Saruman in the afternoon of T.A. 3019 March 5 (S.R. 1419 Rethe 5), Gandalf and company begin their march back to Helm’s Deep.
The road passed slowly, winding down the valley. Now further, and now nearer Isen flowed in its stony bed. Night came down from the mountains. All the mists were gone. A chill wind blew. The moon, now waxing round, filled the eastern sky with a pale cold sheen. The shoulders of the mountain to their right sloped down to bare hills. The wide plains opened grey before them.
J.R.R. Tolkien, The Two Towers, “The Palantír“
That night they made camp at Dol Baran, west of the highway along the Isen.
This is also the day that Frodo, Sam, and Gollum hide in sight of the Morannon, and then they leave at dusk.
The dusk was deep when at length they set out, creeping over the westward rim of the dell, and fading like ghosts into the broken country on the borders of the road. The moon was now three nights from the full, but it did not climb over the mountains until nearly midnight, and the early night was very dark. A single red light burned high up in the Towers of the Teeth, but otherwise no sign could be seen or heard of the sleepless watch on the Morannon.
J.R.R. Tolkien, The Two Towers, “Of Herbs and Stewed Rabbit“
This was the night of March 5/6 (Rethe 5/6), and the full moon is the night of 7/8, so I would think that means the moon on this night was only 2 nights from the full. The quote above is from my 1965 edition, and my 2005 copy of Hammond & Scull’s The Lord of the Rings: A Reader’s Companion does not have a note on this passage. According to my Google searches, it appears this still reads as “three nights from the full” in the 50th anniversary edition.
On this night, the Moon would reach its zenith in the south just a couple of hours before midnight, and I would expect it to rise over the Mountains of Shadow a bit earlier than that.
Back at the camp at Dol Baran, while everyone else was sleeping, Pippin continued to obsess over the Palantír he picked up back at Orthanc, which Gandalf confiscated from him.
At last he could stand it no longer. He got up and looked round. It was chilly, and he wrapped his cloak about him. The moon was shining cold and white, down into the dell, and the shadows of the bushes were black.
J.R.R. Tolkien, The Two Towers, “The Palantír“
Pippin was able to sneak the Palantír away from sleeping Gandalf.
He stole away, and sat down on a green hillock not far from his bed. The moon looked in over the edge of the dell.
Pippin sat with his knees drawn up and the ball between them. He bent low over it, looking like a greedy child stooping over a bowl of food, in a corner away from others. He drew his cloak aside and gazed at it. The air seemed still and tense about him. At first the globe was dark, black as jet, with the moonlight gleaming on its surface. Then there came a faint glow and stir in the heart of it, and it held his eyes, so that now he could not look away.
J.R.R. Tolkien, The Two Towers, “The Palantír“
According to Tolkien’s time scheme, as quoted by Hammond & Scull in their Reader’s Companion, these events with the palantír take place just before midnight. So the moon would have been just past its zenith in the southern sky, making its way west, when Pippin looked into the palantír.
Pippin was then seen by Sauron, who questioned him, and when it was over he screamed from the horror, which woke up everyone else. They feared Sauron knew where they were, but Gandalf assures them that “The Enemy, it is clear, thought that the Stone was in Orthanc… We must snatch that time.” He then advises them to leave the area of Isengard.
At that moment a shadow fell over them. The bright moonlight seemed to be suddenly cut off. Several of the Riders cried out, and crouched, holding their arms above their heads, as if to ward off a blow from above: a blind fear and a deadly cold fell on them. Cowering they looked up. A vast winged shape passed over the moon like a black cloud. It wheeled and went north, flying at a speed greater than any wind of Middle-earth. The stars fainted before it. It was gone…
‘Nazgûl!’ he cried. ‘The messenger of Mordor. The storm is coming. The Nazgûl have crossed the River! Ride, ride! Wait not for the dawn! Let not the swift wait for the slow! Ride!’
J.R.R. Tolkien, The Two Towers, “The Palantír“
Gandalf then gathers Pippin and they mount Shadowfax, whose “flowing tail flicked in the moonlight” before they rode off for Minas Tirith “like the north wind from the mountains”.
Later that night, as the rest of the company are nearing Helm’s Deep once again, they are overtaken by the Grey Company in the early hours before dawn.
The sinking moon was obscured by a great sailing cloud, but suddenly it rode out clear again. Then they all heard the sound of hoofs, and at the same moment they saw dark shapes coming swiftly on the path from the fords. The moonlight glinted here and there on the points of spears. The number of the pursuers could not be told, but they seemed no fewer than the king’s escort, at the least…
The pursuers brought their steeds to a sudden stand. A silence followed; and then in the moonlight, a horseman could be seen dismounting and walking slowly forward. His hand showed white as he held it up, palm outward, in token of peace…
J.R.R. Tolkien, The Return of the King, “The Passing of the Grey Company“
The moon would have set 5 or 6 hours after midnight, so they must have met the Grey Company before then.
By my reckoning, there is more text covering the events on this date in the story than for any other date, since it includes all of Book III Chapters 9 and 10, along with some of Chapter 8 and most of Chapter 11; all of Book IV Chapter 3 with some of Chapters 2 and 4; and the start of Book V Chapter 2. In my 1965 3-volume edition, that’s about 60 pages or so. The date that’s a close 2nd would be for the upcoming date of the Battle of the Pelennor Fields on T.A. 3019 March 15 (S.R. 1419 Rethe 15), which gets about 50 pages or so, in comparison.
The Shire Reckoning Project 