In Harry Potter and the Cursed Child and many quotes from J.K. Rowling, I think Hermione Granger starts to work at the Ministry in the Department for the Regulation and Control of Magical Creatures, becomes the deputy Head of the Department of Magical Law Enforcement, and eventually becomes Minister of Magic. Is there any clue to when Hermione starts to work at the Ministry?
1999 probably, based on a couple of quotes from JKR about Hermione's time after Hogwarts.
We know she completed her N.E.W.T.S. We can assume this was in 1998 after school restarted since there's no mention of her taking a break
JKR: No, she would definitely, definitely go back. And she would want to graduate, and I think that she was- I mean,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.
and that she "began" her career afterwards, within the Ministry, again with no mention of her taking a break.
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.