There was a proto-Austrian line of thought among classical economists. It started with Richard Cantillon went through Turgot to J.B. Say and the French Liberal School, including Frederic Bastiat. This line was part of what is called the continental classical economists. Bastiat influenced the leading American economists before the end of the 19th century including Francis Wayland.
The British classical economists, Adam Smith, David Hume, David Ricardo, Jeremy Bentham, Robert Malthus, James Mill, John Stuart Mill, were the forerunners of today’s neoclassical school.
Take a look at Murray Rothbard’s second volume on the history of economic thought, Classical Economics, to get an idea of which classical economists would be valuable to read. His discussion of Cantillon, Turgot, and Smith are in the the first volume of his history of thought, also available online at mises.org.