The most I've ever been able to obtain my b/p's for @ T-24, prior to the first flight is 3 segments. I have always needed to wait until T-24 prior to the 4th segment in order to check-in for that last flight.

Disclaimer --- I usually only do 4 segments per day on MR's and never do more as the OP indicates he is doing. However, 3 b/p's are all that are ever available on the first encounter @ T-24 prior to flight #1 in my experience.