when you have correct name of his grandma - the death-date you have - so for that time ask at Standesamt Salzburg. would not state the adoption. as more then 50 years when she died so no problem to get a document about that normally, also without a proof for relationship. write only you like to get her death-document and ask also for death-document of the grandpa who died at accident. were they married? if not do you know his name?
does your husband have a birth-document of his father with mothers dates (birth) and his fathers dates too. if was illegitim father will not be stated (normally). but guess he will not have. :(
hope belonged to town Salzburg where she died . but you will see if they find her death in the files. the e-mail-adress on end of page: http://www.stadt-salzburg.at/internet/leben_in_salzburg/gese...
if all works so you have then her birthdate and maybe more. before 1938 you always must search then at diocesan-archive. there you can get more to the ancestors. when she also was born in Salzburg here their archive. e-mail-adress also on end of page: http://www.kirchen.net/archiv/
maybe you tell if all worked correct, so we know where to do the further research. also then at parish-church directly his grandma was born would be a possibility.