Estimation of parameters in stochastic processes has been thoroughly investigated for decades and the asymptotic properties of the estimators are known. However, reaching the regime where the asymptotic properties are valid might require such a long time that the well based theoretical results have no practical value. One example of this situation is at the center of our interest. It concerns determination of the time constant in stochastic Langevin equations with additive or multiplicative white-noise terms. Often the number of observations to achieve the asymptotic conditions is beyond the physical limits. Here we show how to overcome the problem by external perturbation of the system. Furthermore, we show that the perturbation is not at the price of deterioration of the estimates of other parameters unless the observation interval is very short compared to the typical time constant of the system. Three processes from the class of Pearson diffusions are studied. They are frequently used in many applications, in particular, they are examples of leaky integrate-and-fire models, which describe the electrical properties of a neuronal membrane. These neuronal models are often used as examples of systems with excitable dynamics. The most commonly investigated process is the Ornstein-Uhlenbeck process, which has additive noise. Furthermore, the square-root process and the Jacobi process are examples of processes with multiplicative noise. The results are illustrated on computer experiments, which show a striking improvement of the estimates of the rate parameter. It has implications for experimental design, where the information about the parameters can be increased for the same amount and cost of data, which is particularly important when samples are expensive or difficult to obtain.