We were talking, and she told me that she had lost her job last week.
Use a comma to indicate that this is a compound sentence.This a sentence that uses past tense and also refers to actions in the more distant past. Therefore, "had" is correct.
We were talking, and she told me that she (had) just lost her job last week.,
Again, use a comma to indicate that this is a compound sentence.Again, this is a sentence that uses past tense and then refers to actions in the more distant past. Therefore, you need "had" and the word "just" modifies "lost," emphasizing the loss.

Olya T.
07/02/24
Ethan B.
Thanks!06/26/24