Again, we continue the story in Skáldskaparmál 17 (Brodeur translation):

On the way to Geirröd's, Thor must wade through the river Vimur, the greatest river of all. He is able to accomplish this task by the help of a staff called gríðarvölr and a mighty girdle called megingjörð, both of which he obtained from the giantess Gridr. During this crossing, Loki is said to hang on the belt of Thor. Thor climbs out of the river by taking hold of a rowan tree. Thus there is a saying that rowan is Thor’s deliverance.

I assume this story is based on star lore. It is written e.g. in the Phainomena of Aratus (270 BC), that  the celestial river Eridanus is closely connected to the constellation Orion. Both constellations are rising together on the eastern horizon. As Orion rises, laying on his side, it indeed looks like the river reaches his shoulder stars. Orion is famous for its belt, formed by three bright stars: Alnitak, Alnilam and Mintaka. Loki is said to hold onto the belt of Thor. I assume this is a reference to Sirius, the brightest star of the sky, rising just after Orion. The girdle stars are pointing exactly in the direction of Sirius. In his hand Thor holds the gríðarvölr. The first stars rising from the constellation Orion is a series numbered the Pi-stars arranged in the shape of a bow or a staff. Thor is said to throw a stone to the source of the river— the source being one of Geirröd's daughters, standing astride the waters and causing them to rise.  This might either be a reference to the bright star Rigel being at the same time the left foot of Orion and the first star of Eridanus or to the star Theta Eridani, also called Acamar (from Arabian Ākir an-nahr, end of the river). 

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But what about the rowan? To solve this riddle we have to evaluate the stars above Sirius. We find here the constellation Canis Minor with the bright star Procyon. Interestingly this constellation is described according to Richard Allen (Star names) in Arabian sources not only as a dog but also as Al Jummaiza, the mulberry tree! It seems that the rowan is the northern equivalent to the Arabic mulberry tree. The meaning could be that starting from the rising of Procyon the constellation Orion is higher in the sky than Eridanus.

But is it really possible that Orion represents Thor in this story? It is said that Thor spent the night with the giantess Gridr and she gave him her staff, gloves and girdle. Here, I think the meaning behind this is that Thor, as the planet Jupiter, reaches Orion (Gridr) and takes “possession” of her. This explains why Thor can be  Orion in this story but as Taurus in other tales. We find this pattern also for Loki (most probably the planet Saturn) taking the shape of a fish, a bird, a horse, etc., all of them references to constellations he “occupies”. 

"Other stars, mean in size and feeble in splendour, wheel between the Rudder of Argo and Cetus, and beneath the grey Hare's sides they are set without a name. For they are not set like the limbs of a fashioned figure, such as, many in number, fare in order along their constant paths, as the years are fulfilled stars, which someone of the men that are no more noted and marked how to group in figures and call all by a single name. For it had passed his skill to know each single star or name them one by one. Many are they on every hand and of many the magnitudes and colours are the same, while all go circling round. Wherefore he deemed fit to group the stars in companies, so that in order, set each by other, they might form figures. Hence the constellations got their names, and now no longer does any star rise a marvel from beneath the horizon. Now the other stars are grouped in clear figures and brightly shine, but those beneath the hunted Hare are all clad in mist and nameless in their course."

After the miserable failure and defeat of his daughters, Geirröd himself attacks Thor. The details of the fight between Thor and Geirröd are so obscure that again, they can only be explained convincingly by using star lore:

 I assume that this bright red star Antares is the glowing piece of iron we are looking for. Geirröd is throwing it against Thor who catches it with the Járngreipr. As I showed before the Járngreipr is identical to the head of the constellation Cetus forming a fist. Now Geirrödr is seeking protection behind an iron pillar, the Járnsúlu. Looking to the region around Scorpion we understand what the great pillar is: It is the Milky Way forming the barrier between the world of the gods and the giants. As the Milky Way is shiny but not as bright as the constellations who often are described as consisting of gold (Gullfaxi, Gullinborsti, Gullinkambi etc.) it's said to be made out of iron. I assume that also the Járnvidr mentioned in Völuspá refers for the same reason to the Milky Way. But where does the hot piece of iron fly to? The answer might be another bright red star: the star Arcturus sticking in the lower parts (below the girdle!) of the constellation Böotes. And indeed I already came  to the conclusion several times that Böotes represents a giant (e.g. Gymir) so he would be a perfect candidate for the identity of Geirröd.

If we summarize the myth of the fight of Thor and Geirröd it describes that at the setting of the constellation of the Scales and Scorpion (Antares) the head of Cetus is rising. And a short time after the rising of the head of Cetus indeed Arcturus and Böotes are setting on the western horizon.

This explanation demonstrates once again that we really need to focus on the attributes of the actors. Thor is not the primary focus of the myth but rather his Járngreipr/Iron Gloves (head of Cetus), as in the case of the Hrungnir myth, where the rising of Mjöllnir (the Pleiades) is the key to the story.