For each ring deeper, do you dig out under the bottom ring until it slides down and another can be added to the top of the stack? That sounds terrifying having a whole stack of concrete shifting around above you.
In historic well digging the walls needed to be secured with wooden planks or bricks and such after digging straight down. Huge risk of the walls falling in. This was a problem just generations ago... And all of the family had stories of grandfather's digging wells that collapsed and that the historic approach was dangerous.
With concrete rings, there is no danger of collapsing. And certainly, when placing a ring, noone should be standing inside!
But yes... - It becomes a little claustrophobic when this becomes deep. I stopped at 4.5m and got used to it by the time we came that deep. - I put a pipe in to blow fresh air to the bottom. This helps with the uneasy feeling of a possible bad smell - there is a risk of dropping a bucket on the head. Use a proper rope, a proper knot, a proper bucket made for lifting and stop when the guy above needs a break