Responding to the level of new cases, Mr Abbott last week told bars to close and limited restaurants' indoor seating capacity to 50%.
Several southern and western states have recorded a surge in cases after lockdown restrictions were eased.
Should US case rises worry us?
On Friday, federal officials had announced federal funding for testing sites that was to be cut off June 30 would be extended for 14 days, but on Sunday, Pence said, "We'll be extending that every bit as long as Texas wants us to".
The true death toll from the virus, which first emerged in China late previous year, is widely believed to be significantly higher.
According to NBC News, Pence wore a mask earlier in the day but did not wear one when speaking at the news conference.
The state reported five new virus deaths on Saturday, its lowest reported daily death toll since March 15.
And, additionally, much will depend on who the new cases are - if they are fit young people, then there will not be a similar rise in deaths.
While coronavirus cases are falling in the United Kingdom, they have passed 10 million worldwide.
However, when Texas State Governor Greg Abbott talked of a "swift and very risky turn" in regards to the numbers of new coronavirus cases and that "over just the past few weeks, the daily number of cases have gone from an average of about 2,000, to more than 5,000 a day", more and more residents of Texas started to look to the White House for further guidance. Where you can't maintain social distancing, wearing a mask is just a good idea.
Health and human services secretary Alex Azar, meanwhile, defended the fact that President Donald Trump has rarely worn a mask in public, saying he does not have to follow his own administration's guidance because as a leader of the free world he is tested regularly and is in "very different circumstances than the rest of us".
In contrast, NY and its neighbouring northeastern states, which were initially hit hardest, have reported a fall in cases and are pressing ahead with reopening plans.
"It was therefore inevitable that there would be cluster outbreaks as infections spilled over from communities into places of congregation such as mines, factories, taxis and buses". What studies have shown, however, is that washing hands isn't as helpful as wearing a mask.
Per CNN, members of the choir wore their masks between songs.