By Thursday afternoon, Alberto's remnants were scattered over central Mexico, according to the National Hurricane Center.
But forecasters said heavy rain amounting to several inches was falling inland in Mexico's Tamaulipas, Nuevo Leon and Coahuila states. South Texas was forecast to see diminishing rain.
Immediately after it moved ashore in Tampico, there was initial disappointment at the meagre amount of rain that fell. Showers had been sporadic through the early morning with the sun even breaking through at times.
But inland heavy rain was causing damage in the neighbouring states of Nuevo Leon and Veracruz.
There, civil protection authorities reported four deaths linked to Alberto's rains. They said one man died in the La Silla river in the city of Monterrey, the state capital, and that two minors died from electric shocks in the municipality of Allende. Local media reported that the minors were riding a bicycle in the rain.
A fourth man in the community of El Carmen, Nuevo Leon, was electrocuted when he tried to repair wires in the rain, civil protection said.
Water also washed out a segment of a major highway connecting Monterrey and Saltillo.
To the south in the state of Veracruz, 24 families were left homeless in the state capital of Xalapa after days of heavy rain collapsed a retaining wall onto three buildings.
Alberto had spurred tropical storm warnings covering most of the western Gulf of Mexico's coastline from Texas to Veracruz. The storm made landfall with maximum sustained winds of 75km/h.
As much as 25cm of rain was expected in some parts of northeast Mexico and southern Texas, with even higher isolated totals possible, according to the hurricane centre. Some higher locations in Mexico could see as much as 50cm of rain, which could result in mudslides and flash flooding, especially in the states of Tamaulipas, Coahuila and Nuevo Leon.
Alberto was bringing rains and flooding to the coast of Texas as well.Â
The US National Weather Service said the main hazard for southern coastal Texas is flooding from excess rain. Areas along the Texas coast saw some road flooding and dangerous rip currents Wednesday, and waterspouts were spotted offshore.
