So many bothersome little details that don't add up, unfortunately for the father's part of the story in particular.
Per MSM (KXAN, link below), he (whose FB labels him as "single") tried to call her at noon (or 1:40 or 2pm in later comments - pretty big time change IMO), and it went to VM.
He didn't try to reach her after that, until the school called him at 6pm?
4 to 6 hours of not hearing back between spouses with a newborn - is unusual, as others have pointed out. This is huge all by itself. Excuses (drinking, whatever) or not: Why wasn't he worried that she didn't call him back? Did he keep trying after that, or not even worry about not being able to reach her for 4 to 6 whole hours?
This lapse of concern or action in the afternoon -
in addition to his picking up the child first, then going home, and then lah-dee-dah getting around to calling others about her whereabouts
after he was home is overwhelmingly suspicious and I'd have to think LE is all over his phone/whereabouts that day/excuses/contacts.
Link:
https://www.kxan.com/news/local/austin/mom-and-2-week-old-baby-missing-last-seen-in-south-austin/
View attachment 220651