Air (not exhaust) in radiator.

take the donut back off

Do a Compression Test on the Engine and see if you have a Blown Head Gasket. Chances are that is what is causing the Overflow... Exhaust Pressure is getting into the Cooling system

Check the water control valve . It most likely isn't letting coolant into the core.

White steam/smoke coming out exhaust, oil in radiator, coolant in oil, smell of exhaust coming out the radiator, spark plugs wet with coolant, a coolant pressure test would confirm bad head gasket.

FLUSH RADIATOR, HEATER CORE AND ENGINE BLOCK.ADD 50 / 50 ANTIFREEZE AND WATER ALSO REPLACE RADIATOR PRESSURE CAP.IF TEMPERATURE GAUGE STILL STAY HIGH THE ENGINE COOLANT TEMPERATURE SENSOR COULD BE FAULTY.

Hello marie, yes this summer has been so very hot...Couple of things i will suggest for your S-10. When you said, radiator flush, was the radiator removed & rodded out, or just a simple do it yourself flush? If it's clogged internally, a flush will not clean a 14 year old radiator. I would consider replacing it, if that is the case.
Also, in Texas, you don't need a high temp thermostat, like the orginal.
I would install one that opens at no higher than 180 degrees... Make sure your air conditioning condensor and radiator are clean on the outside as well, no bugs, dirt & debris so air will flow freely through to cool. Make sure the fan clutch is good shape & working properly.
Use a good clean mixture of 50/50 coolant to water filled to capacity. If you do this, your S-10 will run under 205 degrees in the Tx heat. I hope this helps, thanks for using FixYa.

Did you flush the radiator when you replaced thermostat too? Sounds like it's clogged and the coolant is not circulating as it should. It probably needs to be rodded out or replaced by now.
Check your cooling fans for proper operation. Make sure the outside of the radiator & condensor are clean, and not full of bugs & debris...I hope this helps.

CHECK COOLANT LEVEL.IF COOLING FAN TURNING ON COOLANT TEMPERATURE SENSOR WORKING.BECAUSE IT CONTROLS AND TURN ON COOLANT FAN AT A SET TEMPERATURE.SOUND LIKE ENGINE RUNNING TOO HOT. HAVE RADIATOR FLUSHED.REPLACE THERMOSTAT. HAVE ENGINE BLOCK FLUSHED.REPLACE RADIATOR PRESSURE CAP ALSO.SEE WHAT HAPPENS.

Have you changed out the thermostat and checked for exhaust blow by from a head gasket leak? One quick way to check for blow by would be if you start the car with the cap off on the radiator and see if antifreeze begins to rise up in the radiator before the engine warms up. Exhaust leaking into the water jacket will bubble up through the radiator. This hot exhaust gas will cause you to overheat. A stuck thermostat will keep the antifreeze from ever reaching the radiator to cool down also causing you to overheat.

have you flushed out the radiator ? get some radiator flush and flush it out. There could be a blockage in the radiator.