With unlimited time I would think he could have.
I understand why he chose the time he did though, because he knew A. She was there B. She found most of the clues when she was a child.
But yeah, after that you would think he would have found a different time. Unless, it just had access to that one room. If the tesseract they made, only had unlimited time access to that one room, he would have been stuck picking times only she went in the room.
If they did that, it was on purpose. Due to the closed time loop. They knew she figured it out in that room, so the only location that he would have access to is her room. But he would have to decide the time he would message her. It does seem like he would have made more attempts with her as an adult.