The beginning of his consulting relationship with the company coincided with its decision to illegally funnel money to Republicans
The Bloomberg Businessweek article about Newt Gingrich's $1.6 million Freddie Mac payday notes that he first interacted with the company in 1999, when his primary contact was Robert Mitchell Delk, Freddie Mac's chief lobbyist, and that "he was paid a self-renewing, monthly retainer of $25,000 to $30,000."
That gig lasted until 2002.
Interestingly, that was a time of scandal at Freddie Mac.
Let's go to the Associated Press archives:
Freddie Mac was accused of illegally using corporate resources between 2000 and 2003 for 85 fundraisers that collected about $1.7 million for federal candidates. Much of the fundraising benefited members of the House Financial Services Committee, a panel whose decisions can affect Freddie Mac. The fundraisers were organized by then-Freddie Mac lobbyists Robert Mitchell Delk and Clark Camper, who described them to the corporation's board of directors as "political risk management," the FEC [Federal Election Commission] said.
... Freddie Mac had held more than 40 fundraisers for House Financial Services Chairman Michael Oxley, R-Ohio. The FEC also found Freddie Mac officials used staff and resources to raise money from company employees to give to candidates, and that in 2002 the corporation itself gave $150,000 to the Republican Governors Association. The RGA ultimately returned the money.
Due to that illegal contribution and those fundraisers, Freddie Mac ultimately paid a $3.8 million fine, the biggest to date in Federal Election Commission history. It could be coincidence that, just as a powerful former speaker of the House was hired by the organization's chief lobbyist as a highly paid consultant, it began to funnel large amounts of money to his erstwhile Republican colleagues.