In one of my previous questions, I found out one of the reasons Harry and Ron didn't finish their education was because Kingsley wanted them to help 'clean up'.
Did he also ask Hermione? Did she refuse?
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Hermione wasn't interested in a career in law when she was 17/18, as evidenced by her conversation with Rufus Scrimgeour in Deathly Hallows:
‘Are you planning to follow a career in Magical Law, Miss Granger?’ asked Scrimgeour.
‘No I’m not,’ retorted Hermione. ‘I’m hoping to do some good in the world!’
Deathly Hallows - page 105 - Bloomsbury - chapter 7, The Will of Albus Dumbledore
That Ron laughed at this indicates to me that he wouldn't have been too fussed if Kingsley didn't ask Hermione to join the Department of Magical Law Enforcement immediately following the Battle of Hogwarts.
Hermione elected to finish Hogwarts and take her N.E.W.T.s after the battle and Voldemort's defeat.
We don't see a lot of Kingsley in the books unfortunately. But what we do see is a man who is insightful and reads between the lines with remarkable ease. Recall in Order of the Phoenix when he was in Dumbledore's office with Umbridge, Fudge, Dawlish, Harry, Marietta Edgecombe, and Percy -- he was able to discern that he needed to place Marietta under a memory charm in order to save Harry's neck, just by correctly reading the situation. Kingsley was an Order member; I'm guessing the Order discussed mostly Harry, but also Ron and Hermione in the years between Order of the Phoenix and Deathly Hallows. Instead of excluding Hermione from his new Ministry (in particular the Department of Magical Law Enforcement), Kingsley may just have been perceptive enough to understand this about Hermione:
J.K. Rowling: I love Hermione. She went with Ron and Harry because she has a really good heart. That's not about brain. Ultimately, she had a bigger heart than she had a brain and that's saying something for Hermione. But did she- Was she naturally drawn to battle? No, she wasn't. She's not a Bellatrix. She's not a woman who actually wants to be hurting, fighting, killing. Not at all. She would be glad to go back to school, be glad to get back to study, and then would join them at the Ministry.
LEAKY CAULDRON - INTERVIEW WITH J.K. ROWLING - 12.23.07
Hermione did go on to work for the Ministry of Magic, first in the Department for the Regulation and Control of Magical Creatures, then for the Department of Magical Law Enforcement.
J.K. Rowling: Hermione began her post-Hogwarts career at the Department for the Regulation and Control of Magical Creatures where she was instrumental in greatly improving life for house-elves and their ilk. She then moved (despite her jibe to Scrimgeour) to the Department of Magical Law Enforcement where she was a progressive voice who ensured the eradication of oppressive, pro-pureblood laws.
THE LEAKY CAULDRON - WEBCHAT WITH J.K. ROWLING - 7.30.07
So Hermione got there eventually!
Finally, it's entirely possible Kingsley did ask Hermione to join the new Ministry right away and she simply turned the offer down in favor of completing her studies.
Based on the transcript quoted in the answer to your previous question, it is pretty much a dead certainty that Hermione would have been asked by Kingsley Shacklebolt to help restore law and order (along with Harry, Ron, et al).
However, Hermione is more of a traditionalist/"rule follower", and so she apparently chose to go back to school instead.
Note - I'm not claiming that Hermione would allow the existing rules to get in the way of doing "the right thing", only that, if the rules and the right thing are reasonably close, she will tend to follow them.