Veran said that the government was going to "accelerate and simplify our vaccination strategy".
Some 300 vaccination centres would be operational from next week, the minister said, after initially ruling out such centres.
The original plan had been for the first phase of the vaccine rollout, which began on December 27, to focus on nursing home residents and staff. By the end of the first week, France had delivered just over 500 Covid-19 shots.
Over the weekend, the first hospital staff began receiving the vaccine. The government has now added paramedics and health workers to the first target group.
By the end of January, France will have begun vaccinating people aged 75 and above who are living at home, Veran said.
France received an initial 500,000 doses of the vaccine developed by Pfizer Inc and Germany’s BioNTech SE and was due to receive an additional 500,000 per week. It will also get 500,000 doses per month of Moderna’s vaccine once it obtains regulatory approval in Europe and France.