Per The HIll: They (Flynn's partners and a Turkish company which we should assume was a cover for the Turkish government) first engaged in July of 2016 without a contract, and had a contract signed around September time frame (they were exchanging written contract drafts by then--the indictments apparently don't say exactly when they signed. Payment was to be made on an unreported date to Flynn's company, not Flynn directly. He had several partners and employees, and some of the money was an advance for the cost of making a documentary making Turkey look good.
Timing isn't very good. But, I guess Flynn had to keep his day job on the chance that Trump lost as everyone predicted he would.
Also, its not like Flynn pocketed $550k like Frank alleges. The money went to costs and expenses of the business and project first, and then Flynn got a paycheck, I'm sure. The contract was for $600k, but some money was returned to Turkey for "unfulfilled work" according to the AP. It doesn't sound like Flynn led the negotiation, so I doubt he got the lion's share of the money. Note that Flynn's business partners were named Bijan Kian and Ekim Alptekin, two Turks...so, I do wonder if he was more of a figure head in this operation. Flynn's business partners met with "a member of Congress, a congressional staffer and a state government official" to discuss that Gulen character which Erdogan accused of orchestrating the coup. So, members of his business, who were presumably paid, did a lot of work on behalf of Turkey. Some money was returned. Flynn's name was put on an oped supporting Turkey in November which had been drafted by Kian (per the AP)...that is what started the investigation.