Dr. Anthony S. Fauci, the top infectious disease expert in the United States, predicted on Thursday that the daily death toll from the coronavirus would continue to rise for weeks to come, and counseled patience with the vaccination program gearing up across the nation. Hours later, officials across the United States reported yet another daily record for deaths, over 4,000. The total for the pandemic in the U.S. has surpassed 365,400.

In an interview with NPR, Dr. Fauci said the high toll was likely to continue, and was probably a reflection of increased travel and gatherings over the holidays. “We believe things will get worse as we get into January,” he said.

He stressed that it was still possible to “blunt that acceleration” by strictly adhering to public health measures like social distancing and mask wearing. “Now’s not the time to pull back on this,” he warned.

The virus has exacted a punishing toll across the United States. In recent days, Wyoming and Arizona been among states that have recorded high per capita rates of virus deaths, according to a New York Times database. On Thursday, Florida reported more than 19,800 cases, the state’s highest single-day total. Dr. Fauci advised Americans to be patient with the snags and delays that have slowed the vaccination effort and provoked widespread frustration.

So far, at least 5.9 million people in the United States have received a dose of one of the two Covid-19 vaccines that have been authorized for use, according to the Centers for Disease Control and Prevention. That figure is far short of the goal federal officials had set to give at least 20 million people their first shots by the end of December.

Dr. Francis S. Collins, the head of the National Institutes of Health, said Wednesday he was “not totally surprised” that vaccine distribution had gotten off to a “rocky” start.

“We had this remarkable plan that Warp Speed had put in place to have doses ready to go the very next day after the F.D.A. approval, but that’s a lot of logistics,” he told The Washington Post, referring to Operation Warp Speed, the administration’s vaccine development program. “So maybe we shouldn’t be too shocked that it didn’t go like clockwork.”

He also echoed Dr. Fauci’s warning, and said “the next couple of weeks are going to be really critical.”

But the rollout is beginning to accelerate. More people were reported as having received their initial shots in the third week of the vaccine drive than in the first two weeks combined. The C.D.C.’s count rose by 470,000 between Tuesday and Wednesday. and then by another 612,000 between Wednesday and Thursday.

In a letter on Thursday to Alex M. Azar II, the secretary of health and human services, the American Hospital Association called on the federal government to do more to coordinate what is being done in different states and municipalities.

“Hospitals are committed to be a central part of the vaccination effort, but hospitals alone cannot do it, especially as we care for burgeoning numbers of critically ill Covid-19 patients, and struggle to maintain sufficient staffing work to have enough personal protective equipment and other resources,” Richard J. Pollack, chief executive of the association, wrote in the letter.