This technical communique considers a practical scenario where a classical estimation method might have already been implemented on a certain platform when one tries to apply more advanced techniques such as Moving horizon estimation (MHE). We are interested to utilize MHE to upgrade, rather than completely discard, the existing estimation technique. This immediately raises the question how one can improve the estimation performance gradually based on the pre-estimator. To this end, we propose a general methodology which incorporates the pre-estimator with a tuning parameter lambda is an element of [0, 1] into the quadratic cost functions that are usually adopted in MHE. We examine the above idea in two standard MHE frameworks that have been proposed in the existing literature. For both frameworks, when lambda = 0, the proposed strategy exactly matches the existing classical estimator; when the value of lambda is increased, the proposed strategy exhibits a more aggressive normalized forgetting effect towards the old data, thereby increasing the estimation performance gradually. (C) 2018 Elsevier Ltd. All rights reserved.