I haven’t seen any dates when the dossier was being shopped to the press. Articles like this suggest it was during the summer of 2016 and not at the end of campaign.
The first date that appears in the Dossier itself was June 2016. Mother Jones made sure the FBI was aware of it in July 2016.
...So the timeline is: The first dossier report was June 20, Steele approached the FBI near the start of July, and the FBI began its investigation in late July...