Interesting question. What about it going out makes it taste bad? And how long would it take for a cigar to taste bad once it's out? If you're smoking and it goes out for whatever reason; lack of attention, less than perfect roll, perhaps too moist, whatever, then most smokers just relight when needed and smoke on. So what's the difference between that and relighting 5 minutes later, or 10 minutes later, or and hour, or a few hours or a day later? After how much time after going out does it taste bad, or harsh or ashy, and why so?
I smoke cigars after they have gone out from 5 minutes to 24 hours and the results are sometimes good and sometimes not so good. Many times it depends what my expectations are at the moment. If I finish a cigar but feel like smoking a little more, I light a cigar that I did not finish the day before, usually 2.5-3" worth and many times that works out just fine and tastes just fine too. Other times it does not taste so good. I think it tastes worse if I finish a milder or medium smoke and then spark up a leftover med-full bodied smoke. The opposite has better results.
Some smokers put their cigars down for a few hours, or smoke a pipe bowl inbetween, and then swear when they relight their cigar it tastes even better. I remember this discussion here on the site many years ago. I believe it was Larry (ljlemer) who had this experience. Anyhow, if it works for you then why not, and if it does not then pass.