He could potentially not have a will.
If someone applied for probate or letters of administration to take control of the estate - they would have had to of put out a public notice of some sort. This was traditionally done in the local newspaper of the area. For example, in Sydney it was the Sydney Morning Herald.
Check the records of newspapers from 1959 - 1960 that were in the local area in that he died. You will then be able to determine who administered the estate and whether or not there was a will. Bit of work I know, but that's all I can think of unless there's someone else with a better suggestion?