In Appendix A it says:
Thorongil men called him in Gondor, the Eagle of the Star, for he was swift and keen-eyed, and wore a silver star upon his cloak; but no one knew his true name nor in what land he was born. He came to Ecthelion from Rohan, where he had served the King Thengel, but he was not one of the Rohirrim. He was a great leader of men, by land or sea, but he departed into the shadows whence he came, before the days of Ecthelion were ended.
Thorongil often counseled Ecthelion that the strength of the rebels in Umbar was a great peril to Gondor, and a threat to the fiefs of the south that would prove deadly, if Sauron moved to open war. At last he got leave of the Steward and gathered a small fleet, and he came to Umbar unlooked for by night, and there burned a great part of the ships of the Corsairs. He himself overthrew the Captain of the Haven in battle upon the quays, and then he withdrew his fleet with small loss. But when they came back to Pelargir, to men's grief and wonder, he would not return to Minas Tirith, where great honor awaited him.
He sent a message of farewell to Ecthelion, saying: “Other tasks now call me, lord, and much time and many perils must pass, ere I come again to Gondor, if that be my fate.” Though none could guess what those tasks might be, nor what summons he had received, it was known whither he went. For he took boat and crossed over Anduin, and there he said farewell to his companions and went on alone; and when he was last seen his face was towards the Mountains of Shadow.
There was dismay in the City at the departure of Thorongil, and to all men it seemed a great loss, unless it were to Denethor, the son of Ecthelion, a man now ripe for the Stewardship, to which after four years he succeeded on the death of his father.
And Appendix B says:
TA 2957-80: Aragorn undertakes his great journeys and errantries. As Thorongil he serves in disguise both Thengel of Rohan and Ecthelion II of Gondor.
TA 2984: Death of Ecthelion II.
This is quite clear - Aragorn came to Gondor directly from Rohan, he already had knowledge of the South, when he left he was headed towards Mordor, and it was four years before Denethor became Steward. Except for the name of Thorongil (why would the Rohirrim give him a Sindarin name?), everything matches.
However, we get another contradiction in The Tale of Aragorn and Arwen:
He rode in the host of the Rohirrim, and fought for the Lord of Gondor by land and by sea; and then in the hour of victory he passed out of the knowledge of Men of the West, and went alone far into the East and deep into the South, exploring the hearts of Men, both evil and good, and uncovering the plots and devices of the servants of Sauron.
Thus he became at last the most hardy of living Men, skilled in their crafts and lore, and was yet more than they; for he was elven-wise, and there was a light in his eyes that when they were kindled few could endure.
It came to pass that when Aragorn was nine and forty years of age he returned from perils on the dark confines of Mordor, where Sauron now dwelt again and was busy with evil. He was weary and he wished to go back to Rivendell and rest there for a while ere he journeyed into the far countries; and on his way he came to the borders of Lórien and was admitted to the hidden land by the Lady Galadriel.
But Aragorn was grown to full stature of body and mind, and Galadriel bade him cast aside his wayworn raiment, and she clothed him in silver and white, with a cloak of elven-grey and a bright gem on his brow. Then more than any king he appeared, and seemed rather an Elf-lord form the Isles of the West. And thus it was that Arwen first beheld him again after their long parting; and as he came walking towards her under the trees of Caras Galadhon laden with flowers of gold, her choice was made and her doom appointed.
The text is explicit in saying that it was only after all his far journeys that he became "the most hardy of living Men." The text is also explicit in saying that it was him being "grown to full stature of body and mind" and looking like "an Elf-lord form the Isles of the West," (instead of a man) that made Arwen fall for him. He has to have taken all those journeys by the time he re-meets Arwen.
The tale also contradicts itself. The first part states he did all his journeying before re-meeting Arwen (indeed, that is the entire point of the passage); the second parts states he has not gone into the far countries; and the third part is back to enforcing the point about him being fully grown, and thus worthy of Arwen.
While there is no way to reconcile the three, the plot requires it all to have been done before meeting Arwen again. The first way, covered in The Stewards, makes the most narrative sense; ie. far east/south -> Rohan -> Gondor -> Mordor -> Lothlórien.
As for the quote of "Aragorn went forth again to danger and toil.", the North was no safe place to be. We know that he captures Gollum in the Dead Marshes - clearly he continues to travel. But there is no proof of him far south/east again.
This is reenforced in the paragraph about Gilrean. The text says “and she seldom saw her son again, for he spent many years in far countries. But on a time, when Aragorn had returned to the North, he came to her”.
It is contrasting his years in the far countries with his years in the North. "On a time" is specific. On a specific time, Aragorn came to his mother. The unusual part is "when Aragorn had returned". Look at it this way. Had means already done. It's past tense. If we take it out, the sentence reads, "But on a time, when Aragorn returned to the North, he came to her" That is grammatically correct. On a specific time, Aragorn returns to the North, and goes to his mother. There is no need for the word "had". The only reason to have "had" there is to change the meaning of the sentence: On a specific time, after Aragorn returned already to the North, he visited her. "Had" completely changes the meaning of the sentence.