  The main aim of this paper is to further develop a multiple-correction method
formulated in a previous work~\cite{CXY}. As its applications, we find a kind
of hybrid-type finite continued fraction approximations in two cases of Landau
constants and Lebesgue constants. In addition, we refine the previous results
of Lu~\cite{Lu} and Xu and You~\cite{XY} for the Euler-Mascheroni constant.
