What, then, is the relation of a Third Way so-conceived to libertarianism? I think that Bill Clinton's comment, "the era of Big Government is over" provides the most important clue here. Third Way thinking opposes a further long-term slide toward socialist levels of public expenditure, even as it perhaps does not oppose past movement in a socialist direction. Third Way thinking also aims at innovation that makes use of the free market (capitalist exchange) and at representing the spirit of individualist liberalism. As such, the Third Way involves a moment of libertarianism, perhaps more of the CATO-style than of more strident and purist forms.
Two factors might be motivating this moment of libertarianism. One is a simple ascension to political reality: we are not going to achieve the pure, Nozickean libertarian situation anytime soon, so one might as well offer guidance to the government on how it is to spend our money. And guidance that emphasizes education and health spending, along with historical respect for individual liberties, is preferable to guidance that emphasizes the need for new armor divisions and spying powers.
But a second motive is possible: one might take libertarianism, and, in particular, property-rights anarchism, to represent an ideal for which we are not yet ready. And so one might wish to move toward the libertarian situation (relative to some hypothetical socialist state) without going all the way just yet. I think this first view, regarding the currently non-practical nature of libertarianism, must be the view of any true Third Way thinker. The Third Way is not merely claiming that social spending--e.g., for public education and housing--is preferable to military spending, but that the restraint on the market represented by such spending is needed for the present moment. Some might further be drawn away from more socialist ideologies by an interest in partially achieving the libertarian situation here and now (but only partly).
I would suggest that a situation of anarchy does indeed represent an ideal, the way in which communism represented a superior stage in history for Marx. One cannot move to the ideal directly, but must pass through the necessary stages. These would be Feudalism, Classical Liberal Capitalism, Restrained Free Market (Third Way), Classical Liberal Free Market, and finally, anarchy (the dismantling of government in totum). The Libertarian Third Way thinker is then claiming that we have not properly achieved "Restrained Free Market" and must stay in this stage until we reach a level of culture such that, for example, private charity really would cover the educational and housing needs of civilization.
As to why anarchy: it is a given that less coercion is better than more, all things being equal. As such, if one can have a situation free from private coercion without the public coercion of the police and military, that is to be preferred. If one can also have this situation be equal or better than one with public coercion when it comes to issues of support for the less-well-off, then that is an ideal which does not seem matchable by government. Perhaps this is an un-attainable ideal, and perhaps one can admit to this (breaking somewhat the analogy with Marxism) as a Third Way libertarian, but this does not mean the ideal is without theoretical or political interest.
Does this ideal of anarchy--de facto, an ideal of as little government "as possible"--already inform many Third Way thinkers? I believe so. But it might be useful for them to consider the situation more consciously.