Hagrid's story seems to be unchanged, and Harry is aware of it - he was told he was responsible for getting the conviction overturned and the wand returned. The point is more that Lucius Malfoy doesn't directly control the Wizengamot. His main tool at this trial seems to be rhetoric, drumming up righteous indignation and playing the part of the aggrieved Noble. If Harry stops focusing on Lucius and in stead focuses on the individual voters, he can find arguments to sway different sections.

Hagrid's case sets a precedent which makes it obvious the Wizengamot is playing to a double-standard in this case, but he would certainly have to come up with more arguments. Another point he could make is that Hermione had no motive. Another is that her behaviour before the event was completely out of character. He has Hermione right there, and veritaserum on hand, so if he asked her the right questions under veritaserum he could probably find out about the huge chunk of missing time she has in her memory - good evidence that she was psychologically manipulated.