For the next six months the focus of US forces in the gulf was on the Mississippi River.
For the six months after that the focus of US forces in the gulf was on establishing a foothold in Texas but Galveston was considered a hard nut to crack so the landings were attempted first at Sabine unsuccessfully and then at Brownsville successfully; troops were then advanced up the coast getting within about 40 miles of Galveston.
But then the focus shifted back to Louisiana, US troops were drawn off the Texas coast and Grant decided that all that was wanted in Texas was the foothold at Brownsville.
Attention only returned to Texas a year later once the rest of the Confederacy had surrendered.
The Battle of Galveston on 1/1/63 showed the Union how hard it was to hold Galveston once it was captured. Take out or neutralize supporting Union naval vessels in Galveston Bay and the troops on the island are cut off.
Confederate ownership of Galveston gained them little and cost them little.
Retaking it would gain the Union little and be a constant worry unless is was heavily defended or used as a base for operations in the Austin/Galveston/Houston area. Since operations in Texas made little difference to the big war, it was too hard to justify the large force investment necessary to take and hold the city or area.