Heavy rain is likely to lead to "life-threatening" flash flooding in southern Texas and north-eastern Mexico even as Hurricane Hanna weakens, US officials have warned.
The hurricane made landfall on Saturday but has since been downgraded to a tropical storm.
But the US National Hurricane Centre (NHC) says that rains and strong winds "remain a threat".
Texas Governor Greg Abbott had issued a disaster declaration for 32 counties.
He said coronavirus would complicate the work of emergency services, although a storm surge warning on the Texan coast was later cancelled by the NHC.
Hanna was initially classified as a Category One hurricane, the lowest level on the five-step Saffir-Simpson scale, before being downgraded.
It made landfall on Padre Island on Saturday, and on Sunday moved into Mexico.
With wind gusts of up to 90mph (145km/h), the storm was earlier tearing roofs off homes.
Hanna hit Texas as the southern state struggled to contain the spread of coronavirus.
More than 380,000 cases have so far been confirmed, with nearly 5,000 deaths.
"Any hurricane is an enormous challenge," Gov Abbott said on Saturday. "This challenge is complicated and made even more severe, seeing that it is sweeping through an area that is the most challenged area in the state for Covid-19."
AT 22:00 local time on Saturday (03:00 GMT Sunday), maximum sustained winds were near 75mph, the NHC said.
It added that "rapid weakening is expected as Hanna moves farther inland" over Texas and into north-eastern Mexico on Sunday.
Separately, Hurricane Douglas - with maximum winds of up to 90mph - was approaching Hawaii in the Pacific, the NHC said, warning of damaging winds, flooding rainfall and dangerously high surf.
US President Donald Trump tweeted that his administration was closely monitoring the storms. "We continue to closely co-ordinate closely with both states," he said.