Garden City is a census-designated place (CDP) in Glasscock County, Texas, United States.It lies near the center of the county, about 27 miles south of Big Spring. Garden City serves as the Glasscock County seat, and at the 2010 Census, had a population of 334.The ZIP code is 79738.arden City was the first Texas community platted by renowned surveyor W. D. Twichell, a Minnesota native. In 1893, Glasscock County was organized, and Garden City vied for the county seat along with two other area communities, New California and Dixie.