We consider a path-following problem in which the goal is to ensure that the error between the output and the geometric path asymptotically is less than a prespecified constant, while guaranteeing output's forward motion along the path and boundedness of all states. For a class of nonlinear systems in which the only input into unstable zero dynamics is system's output and paths satisfying certain geometric condition a solution to this problem was given in [12]. For the same class of systems but under more stringent conditions on the path geometry here we develop a simpler solution to the above problem. We assume here that the path is periodic which allows us to exploit averaging tools to construct an open-loop time-periodic control law for the path parameter instead of a hybrid control law developed in [12].