In a previous article we studied the spectrum of the Zhang-Zagier height . The progress we made stood on an algorithm that produced polynomials with a small height. In this paper we describe a new algorithm that provides even smaller heights. It allows us to find a limit point less than 1.289735 i.e. better than the previous one, namely 1.2916674. After some definitions we detail the principle of the algorithm, the results it gives and the construction that leads to this new limit point.