A hotter thermostat won't help at all. I agree with the others that what you report is not alarming at all. 222 °F is well within the envelope, certainly well below what your engine and coolant can handle. When the engine is working hard dragging a load uphill, you expect it to run warmer than usual.
My TV normally runs 190 °F just puttering around, and the ECU doesn't start taking steps to reduce power to prevent overheating until 235 °F. The "you got a problem" BONG sounds if you get to 245 °F.
Since 222 °F seems hotter than normal to you, it may be that your thermostat is failing and should be replaced. But definitely NOT with a hotter-running thermostat. That would not help. Stick with the current temperature range.