The Australian Government's Office for Women provides policy information for the Executive branch. Among the research resources on their site can be found a report from WESNET (Women's Services Network) (also accessible from WESNET's research page) that includes a history of Australian women's shelters.
On page 50 of WESNET: Women’s Refuges, Shelters, Outreach and
Support Services in Australia we find:

They have come a long way from Elsie, the Sydney squat which claims its place as Australia’s first ‘official’ contemporary white women’s women’s refuge, or Women’s Liberation Halfway House which opened later in 1974 in Melbourne.

So the Halfway House shouldn't have been in existence when Elsie opened.

(Thanks to Astra for the comment that supplies the research for this answer)