A couple of thoughts which may have already been suggested on your other thread. I did read the thread as it was unfolding but can’t recall if any of these have been suggested, or if they will be of any use:
I could not see anyone with the surname Hill on the National Archive of Australia website with this service number - 779746. The search engines does not allow searching by number only. Using Ancestry, there are no results for this service number.
Who was the informant on your father’s death certificate? Who provided the newspapers with this service number? Something to consider is that if only one of these numbers was wrong, then it would not match up with the name. There could be an error by whoever provided the number to the newspaper, or the newspaper themselves.
I believe with some army service numbers, you can get an indication of place and rough date of enlistment. I am not sure if this is also the case with WW2 service numbers but it might be worth looking into.
In your other thread you said that NZ army archives have no record of the service number. Depending on how you asked them to search, it may be worth revisiting this to expand the search? For example if you gave his full name and service number and they could not find a match, then it might be worth you asking if they can search by service number alone, by surname and forename alone etc. I am not sure if they can search via birth date, or just first name, but there are many combinations of search terms which could be used to search for him. This might be something you would have to do in person in the Archives themselves if possible once everything reopens. It is difficult to rely on an archivist who is busy with other things, to do a thorough and exhaustive search.
Once again, apologies if this has already been suggested or if I have missed something.