The seasonal cycle of radiation intensity often causes a marked seasonal cycle in the inhomogeneities (IHs) of observed temperature time series, since a substantial portion of them have direct or indirect connection to radiation changes in the micro-environment of the thermometer. Therefore the magnitudes of temperature IHs tend to be larger in summer than in winter. A new homogenisation method, the Adapted Caussinus – Mestre Algorithm for Networks of Temperature series (ACMANT) has recently been developed which treats appropriately the seasonal changes of IH-sizes in temperature time series. The performance of ACMANT was proved to be among the best methods (together with PRODIGE and MASH) in the efficiency test procedure of COST ES0601 project. A further improved version of the ACMANT is described in this paper. In the new version the ANOVA procedure is applied for correcting inhomogeneities, and with this change the iterations applied in the earlier version have become unnecessary. Some other modifications have also been made, from which the most important one is the new way for estimating the timings of IHs. With these modifications the efficiency of the ACMANT has become even higher, therefore its use is strongly recommended when networks of monthly temperature series from mid- or high geographical latitudes are subjected to homogenisation. The paper presents the main properties and the operation of the new ACMANT.