How did Hedwig know Hagrid wanted to send Harry a note?

Question

In one of the first Chapters in the Philosophers Stone, Hedwig delivers a note from Hagrid to Harry, in which Hagrid invites Harry to visit him in his hut.

How did Hedwig know that Hagrid wanted to send a note to Harry? Wouldn't it have made more sense for Hagrid to send his note using his own owl, or a random owl he could get hold of?

Note: There is a similar question here, about how Hedwig knew that the Dursleys would send Harry a present: How did Hedwig know the Dursleys would have a Christmas present for Harry?
But the answers there point out that Harry sent a note to the Dursleys via Hedwig, and they then replied using Hedwig. This is not what happened here.

Answer 1

Hedwig lived in the Owlery with the rest of the owls at Hogwarts, so any owl Hagrid could have used would have required going to the owlery. So it's likely Hagrid just picked Hedwig since she appears to visit Harry most mornings over breakfast anyway. I think this is the most likely explanation.

Another less likely explanation is that Hedwig just decided to see if Hagrid wanted to send Harry anything. Hedwig does this in Prisoner of Azkaban when she flies to France to pick up Hermione's birthday present to Harry. The only canon explanation I can find for this behaviour is that it's part of her magic as a wizarding owl, and that she is particularly smart for an owl. This is first pointed out by Tom, the owner of the Leaky Cauldron, when she arrives right after Harry despite being sent to the Weasleys while Marge was at Privet Drive.