Seeing the temperature needle hit that critical zone means the cooling system is seriously struggling. That low reservoir reading you found is the most obvious and most likely cause of the immediate spike. When the coolant level is significantly below the minimum line, the system loses its ability to transfer heat effectively, causing temperatures to climb rapidly. But if the level had been fine, then you could be right and it could be a failure of the thermostat or the water pump, either of wich is common on these models. Given what you found, I'd bet on the low volume being the culprit.
Before you consider driving it anywhere else, you have to ensure the system is actually filled to the correct level, not just topped off. Just because it looks full doesn't mean the thermostat or the cap isn't faulty. After you add the coolant, let the car sit for a while and watch the temperature gauge closely. The ideal scenario is that after it warms up and the needle climbs, it settles and doesn't start creeping back toward the red zone. If it immediately starts climbing again after being full, then you definately have a mechanical issue that needs a mechanic. If topping it off and letting it stabilize resolves the heat issue, then you're likely fine for the immediate trip, but you need to address why the reservoir was so low in the first place. Low coolant levels ussually point to a leak somewhere—it could be a hose, the radiator itself, or a drain plug. Don't just assume it was a minor evaporation issue since you just finished a long drive. A leak under the hood is a serious maintenace concern.
Regarding waiting before a long drive, if the engine temperature remains stable and within the green zone after the top-off, you should be good. Just keep an eye on the pressure cap to ensure it seals correctly. I've seen cases where the cap fails and the system pressure drops, leading to rapid boiling even when the coolant looks okay. Keep that in mind before pushing it further.