The same corner is just to the left. You can tell by the fire hydrant, which is still there. It's just a different camera angle to fool you into thinking he's elsewhere in downtown L.A.
The scene was probably filmed on a Sunday morning at sunrise, when traffic would have been all but non-existent. After all, people supposedly had flown because of the approaching Martians.
(Other than the narrator, we have no proof they were actually Martians ...)
This comparison clinches it. In the 1953 shot, we're standing on the southwest corner. We can see the 8th St./Hill St. street signs.
So we've seen shots from at least three of the corners, misleading us into thinking we're running throughout downtown L.A. Waste not.