An improved version of an infeasible full Newton-step interior-point method for linear optimization is considered. In the earlier version, each iteration consisted of one infeasibility step and a few centering steps while in this version each iteration consists of only an infeasibility step. This improvement has been achieved by a much tighter estimate of the proximity measure after an infeasibility step. However, the best iteration bounds known for these types of methods are still achieved. Next, generalizations of the improved method to linear complementarity problems are considered.