If all goes well and there are no disruptions to the supply, one million more people would have received shots by April, the Straits Times reported.
Some 250,000 Singapore residents have been vaccinated to date, with around 110,000 having received their second dose, Yong added.
The aim is to get another one million people to take their first dose of the vaccine by early April, added Education Minister Lawrence Wong, who co-chairs the task force tackling the pandemic with Yong.
"If all goes according to plan, then we can progressively roll out to the other age groups beyond March," said Wong. "We will continue to monitor the supply of vaccine, because that's the critical issue that will enable us to get vaccines out... We will try our best to ensure that we can get these vaccines to Singapore."
The ministers were speaking to reporters at a vaccination centre at Jalan Besar Community Club. The centre began operations on Thursday, and is open daily from 8.30 a.m. to 10 p.m.
The Health Ministry said that Covid-19 vaccination for seniors aged 70 years and above will start on February 22, following a pilot for those living in Ang Mo Kio and Tanjong Pagar.
More than 5,000 seniors from both towns have received their vaccinations.
Over the next three weeks, others in that age group can expect to receive letters inviting them to sign up for vaccination.
Yong noted that Singapore currently has 11 vaccination centres in operation, with three more slated to open next week. More than 30 centres will be up and running by the middle of March, with the full complement of around 40 centres to be in full swing by end April.
He said: "We will watch the progress and if need be, we will open up additional vaccination centres to cater to the needs of the population."
Asked about the recent incident of a 72-year-old who suffered a cardiac arrest after getting his first dose of the Pfizer vaccine, Yong reiterated Tan Tock Seng Hospital's initial assessment - that there is no evidence the vaccine had led to the episode. The man was admitted to the hospital's intensive care unit.
"I want to assure Singaporeans that we do take these very seriously," he stressed. "Whether they are related to vaccinations or not, we want to know if there are any such incidents so that we can investigate in-depth and better understand the situation."
Mr Gan also said that the Government will continue to refine its processes to ensure that vaccination will remain safe.
Before getting vaccinated, people will have to answer questions such as whether they have medical conditions that have compromised their immune systems, or if they have had severe allergies.
Volunteers like 70-year-old Agnes Teo have been going from door to door to explain vaccinations to seniors and help arrange their vaccination appointments.
"I explain vaccinations to them and let them know things like whether or not they should wear long sleeves," said Teo, who speaks English, Mandarin, Malay and a few Chinese dialects.