My impression is that the hunt occurred in the dark in an abandoned city. The intelligences of different species of aliens were closely graded, for example 13.7 compared to 13.5. For some reason I associate the story with another plot in which humanity has fled the Earth but centuries later a ship returns to the "Hall of Departure" on Earth. Any clues or leads appreciated.
This sounds very much like Star Rangers by Andre Norton from 1953.
The Central Control ship Starfire crashes on an unknown planet. They discover native inhabitants on the planet after initially believing there was no intelligent life there.
After some time, they discover an abandoned city. Other people from Central Control planets have settled there.
The crew of the Starfire and the people who have taken refuge in the city are from various planets that individually developed life - there's humans and lizard men and other variants of upright bipedal humanoids.
Common to all of them are telepathic abilities. The abilities are rated by numbers. That's your "13.7 compared to 13.5."
There is a (telepathic and physical) fight (at night, in the rain) in the city between one of the refugees (known as "the Arcturian") and a member of the crew of the Starfire. The Arcturian loses - despite having the higher rating. He was sick (had a fever and hallucinations) during the fight. Besides being a megalomaniac he had also accidentally/negligently infected some of the native inhabitants with whatever virus he had.
In the end, it turns out that the planet is Earth. They do indeed find a "Hall of Departure" from which humans had set forth to colonise the other planets thousands of years before. Earth had become a legend, and no one knew really where it was any more.