We consider a discrete time stochastic model with infinite variance and study the mean estimation problem as in Wang and Ramdas (2023). We refine the Catoni-type confidence sequence (abbr. CS) and use an idea of Bhatt et al. (2022) to achieve notable improvements of some currently existing results for such model. Specifically, for given $α\in (0, 1]$, we assume that there is a known upper bound $ν_α > 0$ for the $(1 + α)$-th central moment of the population distribution that the sample follows. Our findings replicate and `optimize' results in the above references for $α= 1$ (i.e., in models with finite variance) and enhance the results for $α< 1$. Furthermore, by employing the stitching method, we derive an upper bound on the width of the CS as $\mathcal{O} \left(((\log \log t)/t)^{\fracα{1+α}}\right)$ for the shrinking rate as $t$ increases, and $\mathcal{O}(\left(\log (1/δ)\right)^{\frac{α}{1+α}})$ for the growth rate as $δ$ decreases. These bounds are improving upon the bounds found in Wang and Ramdas (2023). Our theoretical results are illustrated by results from a series of simulation experiments. Comparing the performance of our improved $α$-Catoni-type CS with the bound in the above cited paper indicates that our CS achieves tighter width.