Please forgive the intellectual sin, but while I've known many (mostly women, surprisingly) people who've described themself as postfeminist, I have never seen literature, theory, or anything well-sourced on the movement, so I'm going to give some thoughts on it after first having us reflect on the wikipedia page on the topic. Also please forgive the lack of great depth on the topic.
First, I remind you that I am of a gender-role-abolitionist branch of second-wave feminism, but reject or find unimportant most of the feminist theory even from that movement and am generally not that friendly to third-wave feminism. I reject most of the social shaping that third-wavers do as unnecessary and often harmful to feminist goals, I reject the politically-correct discourse some strands would have people adopt, and I feel that a nuanced tolerance is a better ideal between perspectives in society than a mandated/suggested respect.
The first historical mention of postfeminism in the article sounds like gender-role-abolitionism (which as I said I approve of), but it also has little in common with postfeminism as I know it.
The first mention-with-continuity suggests it as an alternate to second-wave feminism starting in the 80s; I identify it as a primarily suburban perspective that aims for formal legal equality and female presence in the workplace, but primarily as an alternate to the life of a wife, and without legal protection against wage or other discrimination. Such a view is broadly compatible with most of the political spectrum in America, focusing most strongly on the first-wave accomplishments/rights and considering the further push to be unnecessary and possibly damaging to the "other path" for women, who remain fairly different from men. Christina Sommers is well-mentioned here as an example of this view; I am not sure if she's ever described herself as postfeminist, but her views reflect something close to the center of that strand.
I generally reject the postfeminist movement. While the first-wave struggle is nearly complete in the United States (although not elsewhere, and it would be wise to aid that struggle elsewhere), the second-wave struggle is not close to being won; the more formal of the second-wave issues may be reasonably solid in law (although it's astounding that the hard-right is pushing back against that right now), but the informal and institutional of these goals have quite a ways to go to be realised. Unlike the postfeminists, I see no need to retain the paved path of traditional female social roles; gender-abolitionists either insist on it being pulled up or open to both genders. Reproductive rights and laws protecting equal wages and structural treatment in the workplace are components in creating workplaces where which of the two genders one is doesn't matter. Sommers' books are particularly problematic because they shield traditional femininity and masculinity as tools of conformity that limit men and women and make them stupid (BE A MAN! BE LADYLIKE!) If the loss of the traditional mindset of boys that Sommers decries is actually happening, let us celebrate.
I am all for new thinking in feminism; third-wave feminism is in my view almost-entirely-rotten, and modern feminist theory consists mostly of adding empty verbiage and harmful goals to real problems, and the cult of universal validation needs to go. The ties to other causes, some valid and some not, has done great harm to the acceptability of feminist movements (socialism, great! feminism, great! socialist-feminist theory? Terrible). We should not be bothered to be labelled culturally-imperialist, nor obligated to approve of every life-choice, and we should be wary of theory, commitments, or ideals that suggest different restrictions/goals on one gender than another or that lead to an ugly society. However, whatever new thing we want to see (be it feminism-without-theory, a new theory, or something else), postfeminism is not it. Undoing the work of the secod wave would structurally trap women back in the kitchen, in bad relationships-of-dependence. Let's rewind back to the second-wave, keep some mild form of sex-positivism, and start again from there instead. The post/pseudo feminism of suburbanites is not redeemable.
And one more thing while I'm on the topic; let's ditch the term of "ally" for social justice. It's confused and demeaning; peple who take part in an activism are not (ideally) doing it on behalf of someone else; they're doing it because they believe it's the right thing to do. It fits their theory, their idea of justice, and they're willing to speak and act on its behalf. That's not being an ally. That's being a member. Social justice is not a herd of the injured and the sympathetic; it's a cause. Whatever your identity, feminism is open to you. Whatever your identity, anti-racism is open to you. Reject being marginalised in your own causes; be a member, not an ally. If somebody tries to marginalise you in your causes, reject it or leave the group. You may be asked to listen; do so, but never accept a greater obligation to do so. Learning to care and learning to listen will dissolve any privilege concerns others have, but those are universal obligations. Movements should not expect help when they collect obedient "allies" who are mostly present to listen; this breeds bad habits and closed-minds and stifles internal variety. Movements collect members. Be a member, or reject the movement.