Going back to the first age, let us not forget that the Ents make one appearance in The Silmarillion. After the Sack of Doriath, when the Dwarves are returning East, we hear of the Ents making an appearance on the slopes of the Blue Mountains.
The Silmarillion: Of the Ruin of Doriath wrote:Thus it came to .pass that when the Dwarves of Nogrod, returning from Menegroth with diminished host, came again to Sarn Athrad, they were assailed by unseen enemies; for as they
climbed up Gelion's banks burdened with the spoils of Doriath, suddenly all the woods were filled with the sound of elven-horns, and shafts sped upon them from every side. There very many of the Dwarves were slain in the first onset; but some escaping from the ambush held together, and fled eastwards towards the mountains. And as they climbed the long slopes beneath Mount Dolmed there came forth the Shepherds of the Trees, and they drove the Dwarves into the shadowy woods of Ered Lindon: whence, it is said, came never one to climb the high passes that led to their homes.
Now, i think there are two important notes here about geography. The first is that the Dwarf Road from Doriath basically ran along the northern end of the River Ascar, the northernmost river of Ossirand (Land of the Seven Rivers).
Given that this is the only appearance of Ents in The Silmarillion, it is possible but not certain that the Ents made their way into Beleriand. However, I think it is VERY safe to assume that all of the land to the EAST of the Blue Mountains (what became "Middle-Earth" after the War of Wrath) was fair and open territory for the Ents and it could be assumed they were more or less everywhere in Middle-Earth. In fact, this is backed up by Treebeard's testimony:
TTT: Treebeard wrote:Aye, aye, there was all one wood once upon a time: from here to the Mountains of Lune, and this was just the East End.
'Those were the broad days! Time was when I could walk and sing all day and hear no more than the echo of my own voice in the hollow hills. The woods were like the woods of Lothlórien. only thicker stronger, younger. And the
smell of the air! I used to spend a week just breathing.'
So what we know as Fangorn in the Third Age was the eastern end of a massive forest that seemingly covered much of Middle Earth, at least from (most likely) Anduin to the Blue Mountains, which is where we saw the Ents come down on the Dwarves in the First Age. I really like this overlaid map
that shows both the Third Age of Middle Earth locales along with Beleriand that got buried under the sea after the War of Wrath.
Interestingly, the area where the Ents are seen in the First Age (near Mount Dolmed in the Blue Mountains) is very close to where the Grey Havens were in the Third Age. Looking at the map, all of Ossirand was buried under water, but the area around the River Ascar (where the Dwarven road ran) became the Gulf of Lune and broke through the Blue Mountains right around where Nogrod was located, which became the future locale of the Grey Havens. This also (potentially) lends some credence to Hal Gamgee's assertion of the "Giant Tree-man" he saw on the North Moors, as the Ents seemingly had roamed the area that became the Shire previously.