‎A Poisson distribution is well used as a standard model for analyzing count data‎. ‎So the Poisson distribution parameter estimation is widely applied in practice‎. ‎Providing accurate confidence intervals for the discrete distribution parameters is very difficult‎. ‎So far‎, ‎many asymptotic confidence intervals for the mean of Poisson distribution is provided‎. ‎It is known that the coverage probability of the confidence interval (L(X),U(X)) is a function of distribution parameter‎. ‎Since Poisson distribution is discrete‎, ‎coverage probability of confidence intervals for Poisson mean has no closed form and the exact calculation of confidence coefficient‎, ‎average coverage probability and maximum coverage probabilities for this intervals‎, ‎is very difficult‎. ‎Methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions with increasing bounds are proposed by Wang (2009)‎. ‎In this paper‎, ‎we consider a situation that the both lower and upper bounds of the confidence interval is increasing‎. ‎In such situations‎, ‎we explore the problem of finding an exact maximum coverage probabilities for confidence intervals of Poisson mean‎. ‎Decision about confidence intervals optimality‎, ‎based on simultaneous evaluation of confidence coefficient‎, ‎average coverage probability and maximum coverage probabilities‎, ‎will be more reliable‎.