I'm more familiar with the Bayeux Tapestry than I am with the chronicle accounts of the battle (The Anglo-Saxon Chronicle and Henry of Huntingdon's Historia Anglorum are the two I see that come up the most on Hastings. Huntingdon's is technically a later work and borrows from the first work quite a lot but he's a Norman so his change in perspective is useful).

In the Bayeux tapestry there is a scene depicting Harold's death. You can see an image of it here although you'll need to zoom in to see it in any real detail. The issue of how he dies arises a lot from a problem of interpretation. Just in case you aren't aware in the Bayeux tapestry you can generally identify someone's faction by the direction they're facing, the Norman's all face right. The last two left-facing figures on the right of the linked image are both possibly Harold. There is text above them that says Harold Rex Interfectus Est which essentially means Here King Harold is Killed. The debate is over the fact that the text spans both scenes, the first is a man with an arrow in his eye and the second is a man being hacked down by a Norman knight. I've read an argument in a paper, I can't recall the name now but I can look it up, that actually both are Harold and he was first shot in the eye but not fatally before being killed by a Knight. I've always been of the opinion that the evidence for him being shot in the eye was stronger (I believe at least one other source mentions it, I want to say it's Henry of Huntingdon but I'm not 100% on that) but I've seen more books claiming otherwise recently. Most of these books have been of the more popular-historical variety, though, where making controversial declarations can help sell more volumes.

Jim Bradbury certainly discusses this topic in The Medieval Archer since the archers at Hastings are a major topic of discussion in the history of medieval archery.

In David C Douglas's William the Conqueror the closing of the battle is described by Douglas as: "In any case, the attackers had been given an opportunity to recover. Duke William doffed his helmet, and having displayed himself to his men as still alive he succeeded in restoring order among them. The issue was, none the less, still in doubt. Harold's position had been weakened, but it was still strong, and both sides were becoming exhausted. It was at this juncture, apparently, that William introduced a new element into his conduct of battle. Hitherto the attacks of his horseman and his footmen had been uncoordinated: now they were to be combined. William, it is said, ordered his archers to shoot from a distance high into the air so that their arrows might fall on the heads of the defenders, and at the same time he sent his weary horseman once again up the hill for yet another attack. This time they were successful. It was perhaps now that Harold himself was killed, and now the defenders were overwhelmed, and the hill position taken. A group of housecarls managed to rally for a while at a spot unsuitable for cavalry in the rear of the main position, and to inflict damage on their pursuers. But there could no longer be any doubt of the outcome. The flight became general and soon turned into slaughter, until at last, as darkness was beginning to fall, the duke called off the pursuit and brought his force back to the hill itself. He encamped for the night amid the carnage."

Douglas cites a lot of stuff in this passage and can provide his sources if interested, although I haven't read them.

In any event I think this supports what Valkine has mentioned as an explanation for the death.

Perhaps in that initial volley Harold was wounded by arrow (which didn't appear to be fatal), but was then killed by the cavalry charge. The reference to the housecarls may or may not have been protecting Harolds body, but I don't see why they would attempt to stand their ground around a fleeing army if they were not protecting something.

Edit for spelling.

According to Bennett al's account of the battle in Fighting Techniques of the Medieval World, William ordered his archers to fire higher toward the end of the battle because the tired English soldiers would have found it more difficult to raise their shields to protect themselves. This fits with Harold receiving an arrow to the face, probably in common with a number of other English casualties.

I've read on this sub regarding ancient/classical fighting tactics, where the two sides would engage for a few minutes and then withdraw for a period of mutual rest. And the fact that the vast majority of casualties occurred after one side broke and the pursuing army would chase and cut down the routed enemy.

My question is - was this tactic still used in the middle ages, ie during the Battle of Hastings? If so, in what context would Harold have been killed? During a route? Would he have stopped fleeing with some kind of retinue for a "last stand" knowing they were defeated? Or during this time would there have been more of a "melee" as depicted in popular media where he could have been killed any time during the battle?