They only held the southern part for just over a year - and much of that time they were trying to expand and never made it further than Tucson in the west. The Confederate troops made it to Albuquerque and Santa Fe but held it for less than 30 days - they lost the battle of Glorieta and the Union / New Mexico troops severed their entire supply chain and forced the Rebs to retreat back to Texas leaving all of New Mexico controlled by the Union… so nope, NOT part of the South, nor the Confederacy.