Should note that Handmaiden's Tale was a coded warning about Islam (Atwood had to slap Christianity onto it because her SJW friends would have turned on her if she wrote a book that denounced Islam as evil) and the Sons of Jacob took over largely by just KILLING the sitting government off, via a false flag act of mass murder involving Congress and the President being gunned down by masked gunmen.
(Atwood leaves it unstated as to who was in power when they took over though)
The TV show tries to fill in the gaps that the Sons of Jacob were ultra conservative Christians on the far right, along with center-right trad wives like Serena Joy who were radicalized by SJWs shooting her and leaving her barren from being shot, conservative environmentalists (one of the running bits of world building that the TV show likes to drop is that Gilead is 100% Green tech centered save for stuff like minor thinks like dry cleaning which is kind of a weird thing to drop on people given how "going green" is the left's thing), and libertarian isolationist economic types.
Also, in a plotline that was aborted in season two (while June was hiding in a major Boston newspaper office building that had been abandoned after all of the employees were executed by the Sons of Jacob early in the coup), the show had June combing through the archives of the newspaper trying to create a timeline of how the Sons of Jacob took over and who were the real masterminds behind it. One of the things she found, before the plot got abandoned, was that there was a recording of how the Sons of Jacobs started out as a faith based initiative under Bush II that tried to clean up the ghettos by creating sanctuary style youth centers in major crime filled cities. June listened to a member of the Sons of Jacob speak highly of the group, how they were "bringing order" to the inner cities and offering kids and teenagers an alternative to gang life.
Basically, imagine a scenario where if Trump loses in 2020, that the GOP goes full cuck to the point that they don't even invite the religious right back into the fold because the GOP "doesn't want them around because they were tainted by their tacit support of Trump" and the religious right, the alt lite, and libertarian party all basically gather together to stage a full-on coup and basically let the fundie christians set up a theocracy so long as the others (the libertarian isolationists and conservative conservationists/environmentalists, and the alt lite get to run the parts of society that the religious right don't care about without much hassle.
