A video of the briefing can be viewed in the video player above.
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President and CEO of TMC Bill McKeon, Chief Medical Officer Dr. David Persse and others answered questions.
As of last month, there was a 45% COVID-19 positivity rate increase at TMC hospitals, health experts said during the briefing.
They said they have been at Phase I full capacity 11 times this month.
The first batch of vaccine doses were reserved for frontline workers.
"We delivered 32,376 shots to our frontline staff. That's about 6,000 shots per day," said McKeon.
The State of Texas announced that the next phase of vaccinations will go to those who are 65 and older, and those who are 16 and older and have a serious health condition or are pregnant.
"From the Memorial Hermann perspective, the process of distributing vaccines has worked well today," Dr. David Callender, the president and CEO of Memorial Hermann Health System, said in regards to administering vaccines to health care workers.
As they move forward to engage others in the community, Callender said it becomes more difficult.
"We're collaborating with colleagues across the greater Houston area and will certainly move forward when the state believes that we're ready to do that," he said.
Some major hospitals in the Houston area struggled to get doses of Pfizer's COVID-19 vaccine.
Just yesterday, Dr. Joseph Varon at United Methodist Medical Center got his first shipment of the Moderna vaccine and was one of the first to get it.
Experts divided people into groups to determine when they get the vaccine.
"We're going to ask the public to have a little dose of patience with this," said Dr. Persse. "We certainly don't want everyone coming to the hospitals to get vaccinated. That would not be a wise move. We don't want long lines forming."
After the 1a group finishes getting the vaccine, the 1b group will be divided into two groups -- the 1b employees and the cancer patients that they care for at MD Anderson.
"When we look at the vaccine supplies that we have now, we don't anticipate that MD Anderson would go and into 1b vaccination until at least next week, possibly not until the new year," said Peter WT Pisters, M.D., president of MD Anderson Cancer Center.
Dr. Persse said they have already discussed other networks outside of hospitals to help with administering vaccines.
For example, CVS announced its pharmacies will help distribute COVID-19 vaccinations to more than 40,000 long-term care facilities across the country.
Two thousand of those facilities are in Texas.
As far as which vaccine to get, there is not one that experts see as more preferred over the other.
"We don't really see material differences in the safety profile or the side effects associated with each of these vaccines," said Dr. Callender.
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Dr. Callender said many health care workers are extremely grateful to them for the vaccine.
"I think as more leaders step up and get vaccinated, I think that will really help us in a number of ways to really help elevate the vaccination rates over time," he said.
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