I went through this with my 95 GSR beater. Last spring my car would run hot on uphills, or in traffic with the AC on. I had a coolant system pressure check done, which was fine-and leak down test which was also good. That left the radiator or the water pump(I already had replaced/removed the thermostat and replaced the cap). I wasn't looking forward to replacing the water pump and a few things indicated it might be working.

After one trip my car ran really hot the next morning. I added water and stopped at my friends house and noticed a stream of water was shooting 6 feet in front of the car through the cowling! So, I replaced the radiator and that was the problem for me-it was blocked somehow.

Other stuff you can look for are blocked/collapsed hoses. I had a 240sx with a blown gasket and it would overheat after about 10 miles, not 1 miles as you are describing. The coolant pressure check will tell you a lot-if its leaking in the head it can't hold pressure