The traveling tournament problem (TTP) is to minimize the total traveling distance of all teams in a double round-robin tournament. In this paper, we focus on TTP-2, in which each team plays at most two consecutive home games and at most two consecutive away games. For the case where the number of teams $n\equiv2$ (mod 4), Zhao and Xiao (2022) presented a $(1+5/n)$-approximation algorithm. This is a randomized algorithm running in $O(n^3)$ time, and its derandomized version runs in $O(n^4)$ time. In this paper, we present a faster deterministic algorithm running in $O(n^3)$ time, with approximation ratio $1+9/n$. This ratio improves the previous approximation ratios of the deterministic algorithms with the same time complexity.