I'm not sure what your question is.
ASTM A588 is a weathering steel. It should be welded using low hydrogen electrodes that have been properly stored to ensure they meet low hydrogen requirements. If the weathering characteristics must be retained because it will be used in the unpainted, exposed condition, you can deposit the last two layers using a nickel bearing electrode, i.e., 8018-C2, or other matching characteristic found in AWS D1.1.
Al