Using Zote
Or any other soap for washing even when "cut" with detergent will give softer results.
Have been saying this for ages now, but don't take my word for it:
The "built in fabric softener" Ivory Snow, Fels and other soap based laundry "detergents" were touting is the residue effects from using just that, soap.
True soaps are made from fats and oils; and regardless of how much rinsing trace amounts of same remain in/on fabrics. Hence the "built in fabric softener" claims.
Remember original fabric softeners were merely emulsions of tallow and or other oils/fats in water. First invented for use in commercial laundries, they spread to domestic use for same reasons; to counter the harsh feeling of fabrics washed in the "new" detergents like Tide. For generations housewives, commercial laundries or anyone else had no need of such a thing because they were using soap for washing laundry. For most part
There are other contributing factors to roughness; pH level, type of surfactants and or builders and so forth; but on whole doing laundry with soap will give you a softer result than most detergents, especially powders built with soda, caustics and Zeolites.
Commercial laundries still do wash certain things in specially built soaps (diapers among one thing ironically), and or have access to soap products marketed as wash additives. There they boost cleaning of oily/greasy fouled laundry and or add softness depending upon dosage.