put me on the list for the very next available flight

When do you ask this? When calling an air company?

When you arrive hoping to get on the flight and there’s no availability.

Поставьте меня на/включите меня в очередь на следующий рейс, на который только есть билеты. ¿
Never happened to hear that. Why don't they just book/buy the flight they need?

That’s the thing. It happens when you need to book a last minute flight. They tell you there are not seats and you ask to be put on standby in case someone cancels or doesn’t show up.

[Customer calling the airline]: I know this is last minute, but I've had a death in the family and I need to take the next flight out to Albuquerque. I know there's a 3pm flight ...
[Employee]: Unfortunately, that flight is booked up. Would you like me to put you on standby for this one and book you on the next one at 5pm? We can always cancel it with no penalty if you make it on the 3pm flight.

Thanks a lot

My pleasure))

Да, очень интересно!

😉