In this paper, we focus on the problem of estimating the reproduction number Rt along an epidemic, as it represents the most used tool to study and control this phenomenon. In particular, we focus on two issues. First, when dealing with the standard estimator for Rt , we consider the use of positive test data as an alternative to the first symptoms ones, that are typically used. We study both theoretically and empirically the relationship between the two Rt sequences obtained. Second, we describe a new estimator for Rt that is not affected by the drawbacks due to the hypothesis of its local constancy assumed in the standard approach. We illustrate the results obtained by applying the proposed methodologies to both real SARS-CoV-2 Italian data, and two types of simulated ones. In both the two cases, we apply specific methods we have developed to reduce both sistematic and random errors affecting the data. Our results show that the Rt along an epidemic can be estimated with some advantages by using the positive test sequence, and that the estimator we propose outperforms the standard one. We hope that these new methodologies could help to study and control epidemics, and in particular the current SARS-CoV-2 pandemic.