When I first introduced 2 large groups together. We had to pick up like 5 off the floor and put them back in the pond. I didn't have a net at the time. They all made it, we were checking on them constantly. I'm guessing it was the males chasing around the females. None have jumped out since.
When we throw food in usually the first one that makes it to the food ends up body surfing underneath all the other koi. So I just keep the water level a little low so the surfing koi don't get pushed out of the pond.
Curious how you are managing that many Koi in 350 gallons. Read about all the filtration you have. How much water change do you do a day or a week. 10% a day, 70% per week ? Seems like a lot of fish for 350 gallons. I totally have KOI Envy!!!