Hi, I'm trying to book an award like this: ZHT-ZRH-FRA-IAH.
Online I get ZRH-FRA-IAH with availability.
as soon as I add ZHT in front, it suggests me ZHT-ZRH-XXX-YYY-IAH but from ZRH on with some other flights, which are much less convenient.

I tried to call since all the flights (seen separately) show availability. But the agent couldn't (didn't want to?) book by segment. ZHT-ZRH showed her no availability at all. At the end I asked her to start only with the ZRH-FRA-IAH itinerary and even then she couldn't find availability for the ZRH-FRA segment. After the call, I checked again online and it is available.

1. First, why do call agent have less available flights/seats than online? Is that possible/true?
2. Why can't the call agent book by segment?
3. Is there any workaround for "complexe" booking, like by e-mail?
4. I'm a blue member. That means I only have access to call agents based outside of Europe. I tried the french and the english line; both were unable to book it.

I gladly take any advice on how to book this. Much appreciated.
And sorry if it has been already discussed here. My search in this thread didn't point me at any solutions.