Here's the relevant excerpt from the script of the TV show pilot, for reference:
ZOE: How did you do it?
SIMON: Money. And, and luck. For two years, I couldn't get near her. Then I was contacted by some men, some underground movement. They-they said she was
in danger, that-that the government was... playing with her brain. If I funded
them they could sneak her out in cryo. Get her to Persephone, and from there,
I could take her... wherever.
Technically, this doesn't contradict what we see in the movie.
Simon didn't explicitly say he wasn't a part of the rescue mission himself, only that he was contacted by men who could help him get her out, if he funded their operation. Presumably, based on what the movie shows us, Simon agreed on the condition that he go in with them, the sooner to be reunited with his sister.
As for the cryo, River could just as easily have been put in cryo after they got on the ship. To me it seems obvious that hiding her in a cryo chamber disguised as a crate was mainly to hide her presence in case they were stopped in the longer term, getting off the planet or en route to wherever they were originally going to hide, rather than getting her out of the facility itself (as we might have assumed before the movie showed us otherwise.) Once news of the escape was disseminated, Alliance personnel would be looking for a young girl matching River's description, after all.