I keep literal scripts for phone calls, with branches for the answers I might get. Sharing my basics, please add yours:
Opening any appointment call: "Hi, I'd like to book an appointment with [name]. My name is [name], date of birth [dob]." — front-load everything they always ask, and you skip two exchanges.
When they go off-script: "Sorry, could you say that again? I'm writing it down." — buys thinking time and is entirely true.
Ending: "So to confirm: [repeat everything]. Thank you, bye." — the repeat-back catches errors AND signals the call is over, which is otherwise unknowable.