The definitive source for arrivals in Australia is being developed by the National Archives of Australia at Passenger arrivals index, 1921-1949. As you can see from the title, it does not (yet) include even all those we can confirm leaving the UK.
However, there is a broader name search facility that will run over all immigration and naturalisation records (including those not yet in the Fremantle Index). My quick search suggests there are less than 100 Annie Smiths for you to work through.
Note that all you can see of many records is a one or two sentence summary. But there is a scheme for you to pay to have a record digitised. You get a copy and it becomes available online to everyone else.
EDIT 7 May
Sometimes a wild stab in the dark pays off, although it is not a search strategy to be encouraged.
There was a death registered in Queensland (1978/B89422) in 1978 for Annie Smith the daughter of Thomas Joseph Newton and Alice August. If that is your great grandmother, then she may have migrated directly to Queensland - in which case the Migrant Ship and Aircraft Files held by the Queensland State Archives could show her arrival. I have not checked the files themselves, but can tell you that she does not appear to be in the recently-published index (which you can check on-line).
In the full QSA Catalogue, there are nine migration files for persons called Annie Smith. Given your presumed time window for the migration and that she was widowed, there appears to be just one strong candidate.
Record type: Personal file - nominated immigrants
Description: SMITH, Annie G
Departmental Numbers: 37115/1958
Start Date: 7/4/1958
End Date: 28/1/1959
Be very cautious that Queensland is just one of seven jurisdictions in Australia recording BDM. Given how common the surname Smith is, I would be very surprised if I had hit the jackpot first time. But if you can confirm her mother's name was Alice August...