Running a bit warmer when working hard is completely normal. Just like going for a jog vs walking.
Ultimately if it does approach "over temperature" you need to slow down and continue to downshift to keep the rpm up. If you have a tachometer the RPM should be 1000 to 2000 rpm above the peak torque when running hard up a hill. RPM of peak HP would be about max.
Of course this would be a good time to have the 17 year old cooling system checked and cleaned. Check for debris stuck to the front of the stack that may restrict airflow. Check or just replace the fan clutch (I recommend OEM only). Check or replace the thermostat... seen plenty stuck half open and work fine day to day easy driving.