The term that this is called is "purge". It can also freshen up flavors in a lit stick when you are well in to it or help with a cigar that you smoked too fast and is over heated. I sometimes use it when the wrapper isn't burning right just to get it to even out.
There are people on this board who think nothing of picking up a day old cigar from an ashtray on the patio and lighting up. There are those who say that once it is out, it is out. The purge is a very useful technique, so experiment with it. Also, you can clip the ash end a little bit to get rid of the cold ash that you might inhale when lighting.
I relight cigars all the time when working in the yard. Have no idea if its been 10 minutes or 30 minutes since it went out. Never had a problem.
And I hope that you never do as well
