From the first essay, "Individualism: True and False" comes an interesting thought:
Here I may perhaps mention that only because men are in fact unequal can we treat them equally. If all men were completely equal in their gifts and inclinations, we should have to treat them differently in order to achieve any sort of social organization. Fortunately, they are not equal; and it is only owing to this that the differentiation of functions needs not be determined by the arbitrary decision of some organizing will but that, after creating formal equality of the rules applying in the same manner to all, we can leave each individual to find his own level.
There is all the difference in the world between treating people equally and attempting to make them equal. While the first is the condition of a free society, the second means, as De Tocqueville described it, "a new form of servitude."
(Individualism and the Economic Order p. 14-15)
This strikes me as touching on the sense in which classical liberals in the tradition of Burke and Smith can still be considered "conservative" in the old sense of the term. Although Burke is commonly accepted by those who argue that classical liberalism is not "truly conservative" as being conservative in his outlook because of his reaction to the French Revolution, he was (like Smith) Whig, though they were Old Whigs, not True Whigs or Country Whigs. Prior to the French Revolution, Burke had been generally supportive of the cause of the colonists in the American Revolution.
Taking Hayek's point, classical liberals in the tradition of Burke and Smith do not reject the necessary hierarchy of society. Nor do they embrace sudden, transformative social change. As such, they can certainly be seen as conservative. However, they do seek sufficient freedom within society to allow people to "find their own level", believing that there is a natural hierarchy of ability which will thus result in an ordered society, and a more desirable one than one in which hierarchy comes strictly from birth and rank.
In this sense, the freedom of a classical liberal society creates social order, and a more stable one than the sort that an ancien regime conservatism maintains. Indeed, arguably, at this point in history, it is only this Whig-ish conservatism which is commonly found within society. Ancien regime conservatism has virtually died out.
Entirely different are notions of politics or the human person in which it is held which all people are truly and fully equal -- in ability and inclination as well as in human dignity. Such systems would indeed seem to lead quickly to a most undesirable oppression.