We discuss actions of free groups on the circle with “ping-pong” dynamics, i.e. dynamics determined by a finite amount of combinatorial data. As a natural extension of the work started in (Mann, Rivas, 2016), we show that the free group $F_n$ admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. Following the work of Alvarez, Barrientos, Filimonov, Kleptsyn, Malicet, Meniño, and Triestino, we also exhibit examples of "exotic" isolated points in the space of all circular orders on $F_2$. As an application, we obtain analogous results for the linear orders on the groups $F_n\times \mathbb{Z}$.