In this paper, we suggest novel criteria for pilot designs which minimize outliers for consistent carrier frequency offset (CFO) estimation in the orthogonal frequency division multiplexing (OFDM) systems. In order to derive the criteria, we first analyze the average pairwise error probability (PEP) of integer CFO (iCFO) estimation with consistent pilots in the OFDM systems where fadings among subcarriers are correlated, and address several issues based on the PEP analysis. Especially, the PEP reveals the relation between the channel and the minimum Hamming distance of the training sequence which is composed of consistent pilots. Based on the observations made from the PEP, we revise the conventional criteria to make them suitable for systems for correlated fading channels. Simulation results show that the sequence developed from the proposed criteria yields much reduced outliers compared to the conventional sequences for consistent CFO estimation.