sbiddle:
I don't understand how you weren't booked in at the time. Did you refuse?
It's a requirement to get the 2nd appointment booked at the time of the 1st jab, which will also typically be written on your vaccination card at the same time as well.
The design of the vaccination card would indicate that the plan was to have the second jab booked before you left after having the first jab, but that's not the way it's handled in the Waikato.
I don't have a date for the second jab and was told they would call to make a time, which seems a time wasting way to do it. The best and simplest way would be to book the second one at the time of the first one