By Christina Maxouris, CNN

(CNN) -- Small gatherings are becoming a growing source of Covid-19 spread, a leading health expert said, as cases and hospitalizations are on the rise nationwide.

"In the public square, we're seeing a higher degree of vigilance and mitigation steps in many jurisdictions," US Centers for Disease Control and Prevention Director Dr. Robert Redfield said during a call with the nation's governors on Tuesday. Audio of the call was obtained by CNN.

"But what we're seeing as the increasing threat right now is actually acquisition of infection through small household gatherings," Redfield said. "Particularly with Thanksgiving coming up, we think it's really important to stress the vigilance of these continued mitigation steps in the household setting."

Dr. Jonathan Reiner, professor of medicine at George Washington University, said Wednesday he'd advise people to consider not having indoor Thanksgiving dinners with others who aren't in their immediate household.

"If you're lucky enough to live in a part of the country where the weather will be moderate in November, do an outdoor Thanksgiving. (But) I think in the ... places in the country where the winter comes early, I think you have to really be careful," Reiner told CNN's "New Day."

"The consequences of this virus, particularly for older folks -- the people that we really want to gather with on Thanksgiving -- can be really dire," he said. And frankly, I'd rather do a Zoom Thanksgiving with people that I love than expose them to something that might kill them," he said.

"Next year is going to be much better. Let's get through this, and let's get through it safely."

These comments come as the country's seven-day average of new daily cases surpassed 51,000 Tuesday -- the first time in more than two months that the figure was above 50,000, according to data from Johns Hopkins University.

The average is now more than 48% higher than it was about a month ago, when it dipped to as low as 34,354.

And hospitalizations have been increasing nationwide. More than 36,000 Covid-19 patients were in US hospitals Tuesday, the highest number since the end of August.

White House coronavirus task force coordinator Dr. Deborah Birx gave a warning similar to Redfield's last week, urging Americans to carefully weigh gatherings over Thanksgiving. Even if non-household members appear healthy, it's impossible to know for sure without testing, she said.

Health officials have warned that the spike in cases probably will worsen as cold weather prompts more indoor social gatherings, and that surges could overwhelm the healthcare system and kill thousands in the coming months.

Many states are seeing rise in case rates

As of Tuesday, 36 states had seen their Covid-19 cases rise by more than 10% in the past week versus the week prior, Johns Hopkins data show.

In Utah, Gov. Gary Herbert said the state was "facing its most dire episode yet in this pandemic" and announced a new Covid-19 monitoring system for counties.

"We are utilizing 15.8% of our ICU beds to treat COVID-19 patients, more than double what we were before, and our total ICU utilization is at 69.6%," the governor said in a Tuesday tweet. "This leaves our hospitals precariously close to being unable to treat COVID and non-COVID patients in need of critical care."

In Wisconsin, hospitalizations have nearly tripled in the past month, the governor said, adding that residents should do all they can to help get the virus under control and prevent the state's healthcare system from being overwhelmed.

In Washington state, one public health official warned of a dramatic rise in cases.

"We're having over 140 cases reported each day over the past week," said Dr. Jeff Duchin, health officer for the Seattle and King County public health agency. "This is more than twice what we saw in late September."

Rising cases prompt restrictions

Several local leaders have pushed for new measures following a rise in cases. Oklahoma City leaders extended the local mask ordinance until at least December 7 amid a rise in Covid-19 hospitalizations, health officials said.

Director of the Regional Medical Response System Heather Yazdanipour told CNN the city was currently in a surge environment and hospitals are experiencing shortages in equipment, space and staffing to accommodate the incoming patients.

New Mexico Gov. Michelle Lujan Grisham announced several restrictions Tuesday to help curb the spread of the virus amid an "alarming" rise of cases in the state.

"When the community spread of the virus becomes uncontrollable -- and we are fast approaching that point -- our only option is to simply shut down those opportunities for the virus," the governor said in a statement.

Bars and restaurants serving alcohol will have to close by 10 p.m. starting Friday and mass gatherings will be limited to five people.

"Rollbacks will mean more economic turmoil for so many workers and business owners in our state who have already suffered and sacrificed so much," the governor said. "But it is our only chance to prevent more devastating illness and to save lives."

Fauci: Vaccine development 'on a really good track'

Meanhwile, despite news that a second Covid-19 vaccine trial was put on hold this week, Fauci said during the governors' call Tuesday that efforts to develop a vaccine are on a "really good track."

"A couple of the vaccines are very close to getting some sort of information," he said, according to the audio obtained by CNN.

US health officials should know by November or December whether there is a safe and effective vaccine, Fauci said, adding, "It is conceivable that we might even know before then."

When a vaccine is deemed safe, experts have said it should first be distributed among healthcare workers and vulnerable populations, before eventually becoming widely available to the US public.

During the call, Health and Human Services Secretary Alex Azar reminded states there is an upcoming deadline for them to provide a vaccine distribution plan to the CDC.

"We want you to be in close touch with your CDC liaison throughout this process as you approach the deadline for submitting a vaccine distribution plan to CDC, this Friday, October 16," Azar said.