The Late Acceptance Hill-Climbing (LAHC) algorithm has been recently introduced by Burke and Bykov. It is a simple, general purpose, one-point search metaheuristic that has similarities with Simulated Annealing (SA) in the sense that worsening moves on a current solution can be accepted. One of its advantages relative to Simulated Annealing is that no cooling schedule is required and its sole parameter, the so-called history length, has a more meaningful interpretation from the application point of view and is therefore easier to specify by a user. In this paper we show that even this single parameter can be eliminated, making LAHC simpler to apply in practice. The validity of the method is shown with computational experiments on a number of instances of the Travelling Salesman Problem.