Hayek himself used the term "neo-liberal" to describe the "movement" of liberalism to which he belonged [0]. The historian Quinn Slobodian [1] advocates using the term "neoliberal" for the intellectual history surrounding the Mont Pelerin Society. In this context, it is sensible to distinguish between a Hayekian strand and, say, Wilhelm Röpke's version of neoliberalism.
[0] The Freeman, 1952. https://mises.org/library/freeman-july-1952-b
[1] Globalists. The End of Empire and the Birth of Neoliberalism.