If you have a minor cut in the tire itself, it can be 'bridged' while you are on the road with a dollar bill. Fold the bill a couple times and put it inside the tire (between tube and tire). We usually carry a small piece of nylon cloth in the patchkit for such rare occasions. A regular glue on patch will also do the trick.
Suspect your blowout while standing still was due to a small cut in tire itself and innertube 'pinched' at that spot. If your tire otherwise is still in good shape, it is re-usable if bridged or patched from inside the tire.