THE BAKER ADMINISTRATION accelerated its vaccine rollout, announcing that starting Thursday individuals over 65 and those with two or more underlying health conditions can start booking appointments for initial doses of COVID-19 vaccines.

Last week, the administration had indicated it would be “several weeks” before it would move on from vaccinating those 75 and older. But Marylou Sudders, the secretary of health and human services, said in a briefing with reporters on Wednesday that the uptake by the over-75 group went better than expected, resulting in more than 50 percent of the group receiving at least a first dose.

“We thought it was going to take longer for individuals over the age of 75 to become vaccinated,” she said. “We’ve been really pleasantly surprised and pleased with the uptake.”

It’s unclear whether the Baker administration’s controversial plan to allow those 75 and over to bring a companion, who could also be vaccinated, played any role in the uptake. Sudders called it an effective strategy, but she acknowledged “people either love it or hate it.”

The new group eligible to get vaccinated includes about 1 million people, Sudders estimated. It includes those 65-74 as well as people with two or more conditions making them particularly prone to COVID-19. Sudders said moderate to severe asthma is being added to the list of conditions, which included cancer, chronic kidney disease, COPD, Down syndrome, immune-compromised states, obesity and severe obesity, pregnancy, sickle cell disease, type 2 diabetes, and smoking.  She also said residents and staff of low income and affordable senior housing will also be part of the group.

At a State House press conference, Gov. Charlie Baker said people with underlying health conditions will have to attest that they have them but won’t have to provide any proof. “It is based on an honor system,” he said.

Sudders estimated it will take about a month to vaccinate the group, which means the rest of those in Phase 2 — essential employees such as teachers and grocery, transit, and sanitation workers, as well as those with just one health condition – will have to wait longer to get their chance at the vaccine. President Biden on Tuesday indicated he favored giving teachers higher priority for the vaccine to resume in-person learning.

A total of 850,000 individuals have received a first dose of the vaccines, or about 77 percent of those eligible so far.

The state has been receiving about 110,000 doses a week from the federal government, but Sudders said the weekly shipment is being increased to 139,000 starting next week. Unless the supply of vaccines increases, she said the weekly shipments would not be enough to move swiftly through Phase 2 of the vaccination process. “That 29,000 increase is still not where we need it in order to vaccinate everyone in Massachusetts,” she said.

The Baker administration is also making some changes to the vaccine distribution system to make it simpler to navigate and more accessible. The seven mass vaccination sites, as well as the 13 regional collaboratives, pharmacies, and 20 communities considered high risk for COVID, will be prioritized in the state’s distribution system. He said the goal is to step up the pace of vaccine delivery. “This is about speed,” he said.

Baker said the mass vaccinations sites deliver a high volume of vaccine the fastest, but he said the regional collaboratives made up of various local providers have also done well. He said county collaboratives have led the way, with Berkshire County vaccinating a state-high 15 percent of its population, followed by Barnstable County at 14.2 percent, and Franklin County at 13 percent. The statewide average, according to a New York Times data set, is 13 percent.

Sudders said 69 clinics that cater to residents of specific communities will no longer be receiving the vaccine under the new streamlined distribution process. She also said hospitals and hospital networks, which were cut off last week, will remain on hiatus but could receive vaccines if the state’s shipments increase. Anyone who received an initial dose at those facilities will receive their second dose as well.

The 20 communities considered high risk that will continue to receive the vaccine include Boston, Brockton, Chelsea, Everett, Fall River, Fitchburg, Framingham, Haverhill, Holyoke, Lawrence, Leominster, Lowell, Lynn, Malden, Methuen, New Bedford, Randolph, Revere, Springfield, and Worcester.

Baker at his press conference shifted gears on his vaccination message. After two weeks of taking heat for a chaotic vaccine rollout and promising to do better, Baker shifted course on Wednesday, expressing confidence in the state’s approach and putting more of the onus for the slow rollout on the lack of vaccine supplies. He rattled off numbers about the state’s rising ranking nationally on vaccine rollout, and said the real problem is the lack of vaccines to distribute.

“None of us thinks this program is moving fast enough, but you can’t vaccinate people if you don’t have the vaccine to make it available,” he said.

Baker urged the state’s congressional delegation, which has been criticizing him for his vaccine delivery system, to instead put pressure on the Biden administration to make firm three-to-four-week commitments about vaccine deliveries. He said a firm commitment would allow the state to book appointments three to four weeks in the future and make the process much easier on everyone.

Baker also said the rollout was slow initially because his administration decided to target high-risk populations that in some cases were hard to reach. “It didn’t happen hardly anywhere else,” he said.