Take the name of your homeowner and search for them in the Bexar County Clerk’s Land Records database. You will find that website here: https://bexar.tx.publicsearch.us/
Organize these results by “Recorded Date” and look for hits around the year where the family first appears at that address in the directory.
Hopefully, the search will turn up documents showing from whom the family purchased the land/house and that will then give you another name to search in the land records and back over in the city directories.
It’s been our experience that the records in this database, prior to about 1990, are most successfully searched using Seller/Buyer (Grantor/Grantee) names and not when searching by address or legal description.