At lot of it comes down to luck. Some mornings she'll do it, and some mornings she may have been told that she can't do it that morning.
If you're first in line, or have a really cute Alice outfit that would help.
A few months back I was waiting for Alice to come out and when she did 2 adorable little girls in a stroller who were just walking through Fantasyland called out to say hi to her, super polite, no running or jumping or screaming, just called out and waved hello to her. Alice stopped to talk to them, and I knew it'd happen, she pulled them up and they went for a ride, and then did the meet and greet after, despite them not having been in line earlier. This was for her first set that day, around 9:15.
The other option, would be if you have multiple days in MK to potentially ask the character attendant, explain how much your daughter loves Alice, and that you've heard about how sometimes people get to ride with her and you were wondering how one would go about that. They might be able to help you out right then, or have you come back another day. I'm pretty tight with the Stepmother and one morning as they were walking through Fantasyland before their first set simply asked "how does one get on the carousel with you?" and she said "like this..." and dragged me & my friend off. It's all timing though, only happens before the first set most days.