We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest vector problem in the infinity norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2^O(n) (log 1/epsilon)^O(n)$ which improves upon the (2+1/epsilon)^O(n) running time of the previously best algorithm by Blömer and Naewe. Our algorithm is based on a solution of the following geometric covering problem that is of interest of its own: Given epsilon in (0,1), how many ellipsoids are necessary to cover the cube [-1+epsilon, 1-epsilon]^n such that all ellipsoids are contained in the standard unit cube [-1,1]^n? We provide an almost optimal bound for the case where the ellipsoids are restricted to be axis-parallel. We then apply our covering scheme to a variation of this covering problem where one wants to cover [-1+epsilon,1-epsilon]^n with parallelepipeds that, if scaled by two, are still contained in the unit cube. Thereby, we obtain a method to boost any 2-approximation algorithm for closest-vector in the infinity norm to a (1+epsilon)-approximation algorithm that has the desired running time.