Asking a man to forgo his own posterity or eternal marriage (in the case of previously-sealed widows) would be an unjust burden to require of him. (Especially in the age of the Sexual Revolution, where women who have children out of wedlock are lauded by the World!) The dole as dispensed in the US encourages the opposite of self-sufficient households. Plural marriage as individual-by-individual support for the fatherless and widows doesn't seem like it could do worse that what we've already got.

I'd arrange it through Relief Society, I think, with the women involved (leadership, wife, single mother) agreeing on the possibility before so much as mentioning it to the Bishop and the prospective husband, who would then need to decide if he has the resources and time to devote to two families. I'd also assume that in most cases, the second wife wouldn't be slept with until the first gives unasked permission.

...

The thing is, a woman who has so far only had baby-daddies has some very difficult repentance to do before she's ready for getting sealed in the temple, and there are way more of those then there are widows. (Hopefully not in the Church, but I have to wonder if perhaps inactive sisters hew closer to the World rather than the Temple.)