Downward to the Earth by Robert Silverberg
You can see the elephant-like aliens right on the covert art. The plot synopsis from Wikipedia hits on the main points in your husband's description.
There are elephant-like aliens:
Edmund Gunderson was the Terran administrator of the colony world of Belzagor, and he returns to it after it has gained independence, feeling a sense of guilt for the way he has treated its dominant species, the elephant-like Nildoror, whose animalistic appearance had kept Gunderson from taking them seriously as sentient beings.
The humans interrupt the elephant aliens' pilgrimage:
Gunderson's greatest guilt comes from having denied rebirth to seven Nildoror to make them help him repair flood damage.
The main character takes part in the rebirth ceremony along with the aliens:
Nevertheless, Gunderson dares to subject himself to the rebirth ceremony, which brings him a new understanding of the native creatures and new powers by which he can heal Kurtz and impart his new-found knowledge to others.
I found some quotes from the end of the book, where Gunderson goes into the cavern for the rebirth ceremony. It's a cavern/cave system, full of mist, with multiple chambers just like your husband described.
A heavy blanket of fog sat close upon things now. ... He hng back, posting himself behind a convenient boulder and allowing himself wary peeps at the mouth of the cavern. ... The mist, the constant companion here, approached and clung. ... Gunderson was alone. This was his moment to peer into the cavern, but he could not bring himself to make the inspection. His lungs were choked with mist. ...
He went to the mouth of the cavern. ... Straight ahead, flanked to by rippling wings of stone that were parted like frozen curtains, lay a passageway, wide enough for Gunderson but probbably something of a trial for the bulky nildoror who had earlier come this way.
He went toward it.
The corridor was about twice a nildor's width and three times a nildor's height, rising to a peaked vault in which the fungoids dwelled. It stretched for what seemed an infinite distance into the mountain. Branching off it on both sides, Gunderson saw, were secondary chambers and passages.