WASHINGTON — President Biden, confronting a worrisome new coronavirus variant and a potential winter surge, laid out a pandemic strategy on Thursday that includes hundreds of vaccination sites, boosters for all adults, new testing requirements for international travelers and free at-home tests.

After nearly a year of pushing vaccination as the way out of the pandemic, Mr. Biden has been unable to overcome resistance to the shots in red states and rural areas. His new strategy shifts away from a near-singular focus on vaccination and places a fresh emphasis on testing — a tacit acknowledgment by the White House that vaccination is not enough to end the worst public health crisis in a century.