So that we all start on the same page, the popularized notion of "the end of history" comes from the liberal democratic modernist par excellence Francis Fukuyama in an influential paper and then later in even more influential book--though the idea is much older than his usage and it is classically attributed to Hegel and Marx. The basic idea here is that society is progressing toward some ultimate form or pinnacle. For Fukuyama that pinnacle was liberal democratic capitalism.

Baudrillard's use of the idea of "the end of history" is the opposite of this notion. For him, history "ends" because we can no longer truly claim that history is moving us towards progress. The use of the scare quotes by Baudrillard in many passages is intentional to show that he actually does not think history ends in the typically understood sense. In the book Forget Foucault he claims in an interview with Sylvère Lotringer that we are erasing history (and even erasing its end):

“There is no end in the sense that God is dead, or history is dead. I would prefer not to play the role of the lugubrious, thoroughly useless prophet. It is not a tragic event, something highly charged with emotion, something that you could mourn-for there would still be something to be done about it. Suddenly, there is a curve in the road, a turning point. Somewhere, the real scene has been lost, the scene where you had rules for the game and some solid stakes that everybody could rely on.” P.72

And,

“It is true that logic only leads to disenchantment. We can't avoid going a long way with negativity, with nihilism and all. But then don't you think a more exciting world opens up? Not a more reassuring world, but certainly more thrilling, a world where the name of the game remains secret. A world ruled by reversibility and indetermination ...” P.74 (emphasis added)

You are correct in saying that within sociology late modernity and post-modernity are often used interchangeably. German sociologist Ulrich Beck makes this connection explicit in the preface of his book Risk Society: Towards a New Modernity. In the preface Beck alternatively refers to this "new" modernity as trans-modernity and reflexive modernity (a term also used by British sociologist Anthony Giddens).

Lyotard's view is that late modernity and post-modernity cannot be separated. In The Inhuman: Reflections on Time he states, “Modernity is constitutionally and ceaselessly pregnant with its postmodernity” (p. 25). There are also similar passages in the book The Postmodern Condition: A Report on Knowledge. For an example of how modernity itself leads to postmodernity, Lyotard illustrates the breakdown of meta-narratives by talking about the thoroughly modern development of the fact-value distinction in philosophy. For Lyotard, narratives are the stories we create to give meaning to the world beyond just mere facts. However, following Hume, there seems to be no clear way to get from an is (i.e., a positive statement) to an ought (i.e., a prescriptive statement):

"The important thing is not, or not only, to legitimate denotative utterances pertaining to the truth, such as 'The Earth revolves around the Sun,' but rather to legitimate prescriptive utterances pertaining to justice, such as 'Carthage must be destroyed' or 'The minimum wage must be set at x dollars.' In this context, the only role that positive knowledge can play is to inform the practical subject about the reality within which the execution of the prescription is to be inscribed. It allows the subject to circumscribe the executable, or what it is possible to do. But the executory, what should be done, is not within the purview of positive knowledge. It is one thing for an undertaking to be possible and another for it to be just." P.36

Lyotard also has much to say about what he thinks we are moving toward in the book The Inhuman: Reflections on Time. Here he makes the allusion that the biggest obstacle facing humanity is its own death:

”The human race is already in the grip of the necessity of having to evacuate the solar system in 4.5 billion years. It will have been the transitory vehicle for an extremely improbably process of complexificiation. The exodus is already on the agenda. The only chance of success lies in the species adapting itself to the complexity that challenges it. And if the exodus succeeds, what it will have preserved is not the species itself but the ‘most complex monad’ ...” Pp.64-65 His worry is that in trying to extend humanity we will essentially become inhuman in pursuit of the transhuman ideal of living without a body. The point is that even the “most complex monad” of what makes humans human is not complex enough to encompass our Being without a body. Lyotard relies on Heidegger and Dreyfus to make his point: “[H]uman thought doesn’t think in binary mode. It doesn’t work with units of information (bits), but with intuitive, hypothetical configurations. It accepts imprecise, ambiguous data that don’t seem to be selected according to preestablished codes or readability. It doesn’t neglect side effects or marginal aspects of a situation. It isn’t just focused, but lateral too.” P.15

I guess the short answer to all of this is that according to these thinkers there is nothing beyond where we are at.

Just a short disclaimer: By this post I am not advocating that I think these views are either correct or incorrect. I am merely stating what I know and the resources that I know of on the subject.

I hope some of this was helpful!