The federal government previously walked back on several vaccinate targets, including ensuring everyone would have a first dose by October.
Now, the prime minister expects everyone who wants a jab will be able to receive two doses by the year's end.
"By the end of the year, people who have wanted to have the vaccine will have had that opportunity," Mr Morrison told 3AW on Thursday.
But he also said no one could provide a guarantee Australia would be fully open by Christmas because of unpredictability of the Delta strain.
The prime minister encouraged people who had received a shot of AstraZeneca to expedite their second dose.
His advice extends beyond locked-down Sydney to people in other parts of the country.
"If they are in a position to do that, then that is consistent with the license for AstraZeneca," Mr Morrison said.
"I need to get the whole country vaccinated as well and the sooner we get there and it can be done safely then we should."
Mr Morrison will on Friday thrash out modelling with the states to determine what level of vaccination coverage Australia needs to stop rolling lockdowns and border closures.
"Will we get there in one meeting? No, I dont think we will," he said.
Sydney and surrounds will remain locked down until at least the end of August as the city struggles to contain a coronavirus outbreak.
Victoria and South Australia have emerged from lockdown after bringing their respective Delta outbreaks under control.
About 17 per cent of Australians aged 16 and older are fully vaccinated, leaving the country well behind similar nations five months into the rollout.
Disaster support payments have been increased from $600 to $750 for people who have lost more than 20 hours of work a week in lockdown areas.
People who lose between eight and 20 hours will receive $450, up from $375 a week.
Welfare recipients, previously been excluded from the payments, can get $200 if they lose eight hours of work a week.
Business support has also been boosted for NSW and Victoria.
In NSW, businesses and non-profits with revenue declines of more than 30 per cent will receive weekly payments of between $1500 and $100,000.
The expanded program's turnover cap has been raised from $50 million to $250 million.
But Mr Morrison flagged the program could change again if required.
AAP - Georgie Moore and Matt Coughlan