How to slip down the slope to polygamy
The arguments in favor of SSM are diverse, but one common one goes something like this: Banning SSM dooms gays and lesbians to a life without love, because the people they could legally marry are people they are categorically not interested in intimacy with. Therefore we need to treat heterosexuals and homosexuals equally by allowing people to marry someone of any gender. We can call this the "orientation model." It's appealing because it's based on asserting equal rights for people on the basis of an inborn and unchangeable identity, and thus seems to parallel the successes of race- and gender-based identity politics.
If the orientation model comes to be the prevailing way that society thinks about why SSM is being legalized, then any advocacy for multiple marriages that wants to build directly on the SSM campaign* would tend to be pulled into an orientation model of its own. This is already happening to some degree. Many people do describe their desire for either polyamory or monogamy as a sort of orientation. We hear poly people describing how they struggled futilely to stop loving other people once in a relationship and even found themselves repeatedly cheating, while poly-aware monogamous people talk about their fundamental desire for exclusivilty and inability to "overcome" jealousy. The argument would then be that some people need multiple partners in something like the same way that a gay person needs a same-sex partner. (I've even seen it argued that as an incrementalist stepping stone from SSM to polygamy we should start by arguing that some bisexual people are unfulfilled unless they can marry "one of each.")
A different model can be seen if we step back and ask why marriage is important in the first place. After all, many people who are in same-sex relationships and/or multiple relationships have no desire to get married (and may even reject marriage advocacy as an agenda item). The reason we have marriage is because sometimes people intertwine their lives in significant ways, and it's important for the legal system to recognize and facilitate that rather than treating the people involved as strangers. All of the benefits that come with marriage are meant to acknowledge the fact of interconnection, from making it easier to take the same last name through hospital visitation and inheritance rights**. We could call this the "support model."
Pretty much every place that is seriously considering allowing SSM has already gotten rid of its sodomy laws, so same-sex couples exist and form deep connections without direct legal sanction. The problem is that the law still treats them as strangers -- taxing them like strangers, handing over important medical decisions to blood relatives rather than one's partner, etc. Likewise, adultery and non-marital sex are no longer crimes in most places. Relationships with multiple partners exist, and are structured in some cases like unofficial marriages. All that is lacking is a recognition and accommodation by the legal system.
I think a "support model" is a stronger basis for advocating both SSM and multiple marriages. It is also more conducive to extending recognition to other relationships that involve mutual intertwining of lives that don't closely resemble traditional marriage. It would avoid tying claims about marriage to claims about a fixed identity. And I think it also more closely resembles the rationale for allowing interracial marriage, a fight which SSM advocates are eager to position themselves as heirs to. Anti-miscegenation efforts didn't posit that there were people who were "interracial-sexual" and would be deprived of their ability to ever find romantic fulfillment if they couldn't marry someone of a different race. Rather, they pointed to existing interracial couples and said that those couples would be broken up or made second-class, damaging the specific bonds of love that people had actually formed with specific partners.
*It is in no way necessary for a successful campaign for multiple marriages to build directly on SSM. For example, FLDS members and pagan polyamorists might coalesce around a religious freedom argument.
**I think there's a good argument that alimony is the marriage benefit that most clearly illustrates the institution's purpose and value -- alimony exists because people alter life plans (e.g. altering their career) and become dependent on each other on the basis of a commitment of mutual support.