The fewer phone calls you make means fewer chances for stuff ups as well.
My experience was to book leg by leg as they became available, though. I had watched the likely flights for a few days or a week before intended travel date to try to get a handle on what becomes available and how long they were lasting. For me the criticial ones (long distance) were disappearing within hours of appearing so thought it safer to get them as soon as.
These were on QF/JL/AT who first release seats around the 350 day mark ( for those lucky to have Status of gold and up), so other carriers like AA weren't in the mix. Naturally something may come up much closer to travel date, but didn't want to take that chance.
I did all the booking in Sept/Oct and didn't get hit with the 5K change fee for adding flights, which the fine print to that offer doesn't say is eligible for the 'free' change. So maybe it is or maybe I had a bit of luck. I don't think there is one 'perfect' way of getting this done, a better web site would help though. I managed to to pretty much get what I wanted and when, a few cities that were not the first choice but places new to us and excited for the journey to commence.
CBR-SYD-JFK
JFK-CMN
CMN-BCN-AMM
AMM-FRA
FRA-NRT
ITM-HND-SYD-CBR
Midnight UTC seems to be when initial availability pops up each day, although I had several contact centre staff they couldn't see the flights for a few hours after that - whether true or user error I don't know.
