Since posting my last message, I found the answer on another forum.
The Admin1 and Admin2 codes for all countries are stored in their own files: 'admin1CodesASCII.txt' and 'admin2Codes.txt'.
The Admin3 and Admin4 codes ARE also available, but are stored within the individual country files. The Admin1 and Admin2 codes are also duplicated in the same location.
To construct a lookup table for Admin3 administrative areas, load the relevant country file then do a query such as:
SELECT NAME, ADMIN3_CODE
FROM my_country_file
WHERE FEATURE_CLASS='A' and FEATURE_CODE='ADM3'