How could Dumbledore hire Hagrid when he wasn't Headmaster?

Question

In Philosopher's Stone, Hagrid tells Harry how he was hired by Dumbledore.

"I was at Hogwarts meself but I — er — got expelled, ter tell yeh the truth. In me third year. They snapped me wand in half an’ everything. But Dumbledore let me stay on as gamekeeper. Great man, Dumbledore.”

But in the next book, we learn through Riddle's diary that Dumbledore wasn't the headmaster when Hagrid was expelled, Armando Dippet was. Every indication throughout the books is that headmasters make the hiring decisions, and as Dumbledore states in Order of the Phoenix, only the headmaster can allow people to continue living in Hogwarts:

"As High Inquisitor you have every right to dismiss my teachers. You do not, however, have the authority to send them away from the castle. I am afraid,' he went on, with a courteous little bow, 'that the power to do that still resides with the Headmaster, and it is my wish that Professor Trelawney continue to live at Hogwarts."

Confusing matters even further is that Goblet of Fire implies that when Mrs. Weasley was in school, there was a completely different gamekeeper.

Mrs. Weasley was intrigued by the Whomping Willow, which had been planted after she had left school, and reminisced at length about the gamekeeper before Hagrid, a man called Ogg.

Mrs. Weasley is older than Harry's parents, but surely she can't be THAT old- Hagrid and Voldemort are older than 70. This whole Hagrid timeline just seems completely off. Is this just another famous example of JK Rowling's math ineptitude, or is there another possibility I'm missing?

Answer 1

Well, the answer is right there in the second book, chapter 16, The Chamber of Secrets. When Harry met Tom Riddle, and he told him about finding the chamber:

"Hagrid's my friend," said Harry, his voice now shaking. "And you framed him, didn't you? I thought you made a mistake, but -"
Riddle laughed his high laugh again.
"It was my word against Hagrid's, Harry. Well, you can imagine how it looked to old Armando Dippet. On the one hand, Tom Riddle, poor but brilliant, parentless but so brave, school prefect, model student... on the other hand, big, blundering Hagrid, in trouble every other week, trying to raise werewolf cubs under his bed, sneaking off to the Forbidden Forest to wrestle trolls...
But I admit, even I was surprised how well the plan worked. I thought someone must realize that Hagrid couldn't possibly be the Heir of Slytherin. It had taken me five whole years to find out everything I could about the Chamber of Secrets and discover the secret entrance... as though Hagrid had the brains, or the power!

And here comes the answer...

"Only the Transfiguration teacher, Dumbledore, seemed to think Hagrid was innocent. He persuaded Dippet to keep Hagrid and train him as gamekeeper. Yes, I think Dumbledore might have guessed.... Dumbledore never seemed to like me as much as the other teachers did...

So, as we see here, Hagrid wasn't gamekeeper at that time, and Dumbledore kept him, but he persuaded Dippet to keep him and train him to become gamekeeper for later years.