The problem you will have in Scotland is that, unlike for England, you cannot search online for births on
www.scotlandspeople.gov.uk with the family name and the mother's maiden name. You can do this type of search from around 1927, but only in person from one of the main genealogy centres...which won't help you here for now I think.
As Carole has suggested, you could try checking for deaths of parents to see whether any siblings of Joe may have reported parents' deaths. I think mother Jenny may have died in Rutherglen in 1940. Images for deaths can be viewed online for that year on Scotlands People (see here
www.scotlandspeople.gov.uk/Content/Help/index.aspx?r=554&413 for what images can be viewed online instantly and what needs to be ordered, due to data protection issues).
One search that you could do on Scotlands People (SP) is to check for births between 1916-36 for Madden in Rutherglen. I can see 25 entries there which you could view for just 1 unit (max 25 entries per search result). If you haven't already, you can view the marriage cert online for Robert and Jenny from 1916. Their parents' names as showing on their marriage will give you hopefully clues as to possible children's names for them.
Monica
