Baker says about 15,000 people are being tested daily for coronavirus in Mass.

Governor Charlie Baker on Tuesday highlighted the state’s increased level of coronavirus testing while renewing his call for people to keep their guard up against the deadly virus.

“The Commonwealth is seeing a steady climb in the number of new individuals tested,” he said at a State House news briefing. “This is obviously good progress.”

Over the past week, he said, 15,000 or more new individuals had been tested per day. That was up, he said, from a seven-day average of individuals being tested of about 9,000 in mid-June and 12,000 in mid-July.

He also said people are taking advantage of “test resources we’ve made readily available across our state” to get repeat tests, noting that the seven-day average of total number of tests has been above 20,000 for the past two weeks, compared with around 11,000 in mid-June and 16,000 in mid-July.

At a separate news conference later Tuesday, Mayor Martin J. Walsh cited encouraging trends in Boston’s COVID-19 numbers but urged residents to remain vigilant and take advantage of the city’s mobile testing program.

“The uptick that we saw in the second half of July has leveled off,” Walsh said during his briefing outside City Hall. “But we’re watching the data very closely.”

Walsh said the positive test rate in Boston for the week ended Aug. 10 was 2.6 percent, down from the prior week’s 2.8 percent.

“Our mobile testing teams have played a role in increasing both the capacity and the access to testing in our neighborhoods,” he said. “We’re offering testing to anyone who wants it, at no cost, and regardless of if you are symptomatic or not.”

Meanwhile, Baker said the state was “continuing to see success” in increasing testing through the Stop the Spread initiative, which offers free testing in 17 municipalities. The program, which began July 10, runs through Sept. 12, and as of last week had performed more than 78,000 tests.

Baker said that three more municipalities, Salem, Holyoke, and Saugus, would be added to the program.

He noted that officials had recently released town-by-town data on the virus’s spread and announced that an interactive color-coded map showing the local risk levels would be online at mass.gov/covidmap. He also said the state was stepping in to help harder-hit communities.

“But enforcement efforts are only part of this solution to fighting the virus,” he said. “People must stay vigilant and continue to do their part to keep their friends, neighbors and family safe. The virus, as we’ve said before, does not stop at any one town line.”

He said that even in less-affected places, “you must keep doing all the things you’ve been doing to stop the spread. We urge people to continue to use common sense and keep doing those things that we know work: Wear face coverings, maintain social distance, and practice appropriate hygiene.”

In other developments:

— Baker offered little hope for those who would like the state to resume its reopening plan, which has been paused at Step 1 of Phase 3. Asked when the state would proceed to Step 2 of Phase 3, Baker said, “I think we need to do some work to deal with the communities that are currently struggling before we go any further down the road.”

— Baker announced $3.5 million in grants to combat food insecurity. The money was distributed under the food security infrastructure grant program.

— The governor said he supported mail-in balloting, which has been in the news because of Postal Service cuts that critics say would jeopardize elections in which many people will be voting by mail because of the pandemic. (The cuts were suspended Tuesday.) Noting that he had signed a bill authorizing voting by mail in Massachusetts, Baker said: “I think it’s critically important, given that there are sort of 11 or 12 states, I think, that are voting, that are going to have big vote-by-mail programs in the fail, that this be done well, and that it be done on a timely basis, and it needs to be part of the way people feel about the election process generally.”

— Baker, a Republican, said the state had written a letter saying it would apply for funding from a controversial program created by executive order of Republican President Trump that would pay a federal unemployment benefit to those out of work during the pandemic. Baker had previously questioned the use of Federal Emergency Management Agency funds for the program, saying the state was expecting the FEMA money for other coronavirus-related expenses. “I don’t think this is the right way to do this,” Baker said Tuesday, “but if this program is there and it turns out to be the only thing that’s there, I don’t think Massachusetts should pass on that.”