To get Polygamy accepted you probably need to make it legal, or at least difficult to ban. You'd probably have to play with the wording of the 1st Amendment on religion, as well as supporting materials. Reynolds vs. United States was unanimous on the grounds of saying that the banning of Polygamy in the United States was constitutional. The argument was that while the government could not touch religious beliefs, it could do the same to religious actions and argued that in principle if any action could be required by religious freedom, then the law would mean nothing. So you need to change the wording of the First Amendment and supporting materials so that it cannot be banned short of a constitutional amendment and that doesn't happen. I'm not sure how to plausibly have such a thing, and how such an amendment would read
Regardless another factor is to have Polygamy challenged earlier, rather than 1878, 1820's or earlier if possible, and a Supreme Court forced by *1st Amendment to legalize it. Have it early enough and more of the groups caught up in the Second Great Awakening accept it, rather than basically just Mormons. You might be able to have William Miller go the same way as Joseph Smith and pick up smaller restorationist groups. It is not much, but it is a start in having a broader base of support
Making it popular, *Civil War that really is a bloodbath by standards of OTL as others mentioned. This makes the practice more widespread, Polygamy being considered a way to support widows and otherwise unmarrigeable women in certain groups. Have an *WWI that is also bloody by European standards to really cement it, as at that time there would still be plenty of living examples. After that it's just tradition, and the Polygamist accepting groups being relatively more successful as they have more kids. Might possibly lead to easier immigration rules for women in order to avoid any "Lost Boys" issues. You could probably also arrange an *WWI wave of immigration from the former OE and bring in Muslim immigrants that accept the practice, and well it's also accepted in East Asia at this time, find was to ensure those people can immigrate and you get more minorities accepting of the practice
It will probably never be something that involves the majority of Americans, and might arguably be considered "Americas Shame" for a good long while, but it will be entrenched