While growth increased in the Texas economy, the oil-rich state still faces significant headwinds because of the energy market downturn, a state economist said.

"Actual employment data for first quarter tell us that the Texas economy didn't grow as fast as we had anticipated," Federal Reserve Bank of Dallas economist Jesus Canas said in a statement. "The combination of slow growth in the U.S. economy during the first quarter and oil prices around the low-$30s heavily impacted the Texas economy."

Texas is the No. 1 oil producer in the country and witnessed financial pressures because of the downturn in crude oil prices. The reserve bank in July said the state's overall economy had improved, though unemployment rates moved slightly higher in June. Its employment forecast for the year showed an expected 0.5 percent increase in employment, down from the previous forecast of 1.3 percent.

Canas said revised data show there are signs of recovery on the horizon, though a review of first-quarter data shows obstacles to growth were more severe than anticipated. Growth improved for the second quarter as crude oil prices recovered somewhat, so the second half of the year should be better than the first, he added.

Despite a price recovery, the Dallas bank noted crude oil prices were still down by about 20 percent year-on-year.

The bank warned earlier this year that the pressure from low oil prices was spilling over to other parts of the economy, with banks in southern U.S. states facing increasing risk.The city of Houston, meanwhile, is facing prolonged headwinds following fits and starts over the last several quarters.

The state is the No. 1 oil producer in the country. The manufacturing sector in Texas is the second largest in the country, behind California, and represents 11 percent of all manufacturing activity in the United States.

"We export a lot of what we manufacture," Canas said. "So if the price of our products is high, the demand for our products drops, forcing some Texas manufacturing companies to cut payrolls in order to compensate."