In my impression, there is one word means the time is almost exactly, like "almost exactly 1 hour".
It is less precise than exactly. (Is "punctually one hour" a right term?)
But more accurate than about 1 hour. (How to say "about one hour"? 1-hour-ish?)
Best Answer
The customary, normal, and most common way to say about one hour in a way to make it more close to exactly one hour is to say just about one hour.
That’s because just here modifies about in a way to make it more precise or more exact, closer to exactly that rather than only loosely or approximately that.
So saying “I’ll be there in just about an hour works to say that I will be there in very close to one exact hour.
You could even say just about one hour exactly, but that’s not really necessary here. The just about suffices.