I am glad you got a good outcome, but would like to answer because I have a fair amount of experience with difficult cooling system issues, and the most recent one occurred while I was over 1,000 miles from home. Had I been close to my home it would have not been very inconveniencing. On top of that, it was the hottest time of the year and I had traveled 1,000 miles south into hotter territory. The issues began about 40 miles from home and I made the very bad decision to top off the coolant and proceed rather than turn around back home and choose a different car. Below is my advice based on a lot of experience with managing a vehicle that is overheating.
First rule is simple -- if the temperature is above the normal (usually half-way, or in some cars just a hair above half-way on the gauge), pull over and stop the engine. That is your safest bet to avoid damage and turning a minor problem into a major one. Most cooling issues are solved by replacing a hose, an $8 thermostat, a $50 radiator fan motor, or a $200 radiator. Water pumps are cheap, but not easy to change on a lot of modern cars, so I don't consider that a "minor" fix because of the high labor costs. Driving the car after it crosses the normal temperature for any amount of time increases the chances that a cheap problem will turn into a $1,00 head gasket or $2,000 engine rebuild.
If for some reason you cannot do that, your first order of business is to turn off the air conditioner, roll down the windows, and turn the heat onto the hottest and the blower to full blast. This will increase the cooling system capacity by allowing the heater core to dissipate engine heat -- obviously at the expense of cabin comfort. Also, by killing the AC, the AC condenser is no longer dumping an additional heat load onto the cooling system. If you do this and the temperature stays within normal range (which is often the case) pull over when you can and check the coolant level.
If the coolant level is down, fill it. If you cannot get proper coolant and have to use water from a jug or a hose, that is better than low coolant. Keep in mind that you should circle back after the repair and drain/flush and refill the system (if it was not required for the fix) so that you have the proper proportion of anti-freeze to water. Also keep in mind that spring water is best for your cooling system, and if you ended up getting it from a hose that might introduce contaminants that over time will cause problems, like corrosion or mineral buildup that clogs the radiator, etc. But, dirty water is better than no water.
If the coolant level was low, the question becomes "how fast is the leak"? If you can't see it, like gushing out of a broken hose or oozing from the radiator, then all you can really do is fill it and then proceed (AC off, heat on, driving very delicately) and see how far you get before the temp breaks the normal reading. Obviously, you carry extra water/coolant with you when you go down this path.
If the coolant level wasn't low, then proceed.
As you drive with a cooling issue you want to avoid stopped traffic and keep the car moving 20mph or better. The reason is the movement creates airflow over your radiator. When you are stopped in traffic, there is no airflow, and the only thing keeping you from overheating is the radiator fan -- and it is very possible that has failed and is the reason you overheated to begin with. If temps stay normal while moving but creeps up over normal while stopped, it is probably the fan. Avoid stopped traffic.
You also want to avoid heavy loads. You may be able to drive with a compromised cooling system at 55mph without overheating, but at 65mph it overheats. This is because engine load is higher, and if you are on the highway driving a long time with a compromised cooling system, the load is higher than the cooling system can dissipate the heat.
Hills are not your friend when driving with a compromised cooling system. If you can't get up the hill without loading the engine to the point it overheats, there isn't much you can do about that.
It is worthwhile to also check the auto transmission fluid, because if that is low the tranny is running hotter than normal and most radiators have a section that is your transmission fluid cooler which is mounted to the radiator putting additional heat on the cooling system.
Using these techniques you might be able to get a car where you need it to be with a compromised cooling system. It is impossible to guarantee results.
If the question becomes "how hot can I run it without doing major damage", the answer to that question depends on the specific engine. Some cars, such as those with steel block and aluminum head, any overheating runs the risk of warping the head (because steel and aluminum expand at different rates). If you can manage to get home or to a shop by using the above techniques, there really isn't too much to worry about. But if you can't do that and must press on with the temp over normal, you are running a serious risk of making a bigger problem. Whether damage is done depends on how hot it got and for how long.