I've tried the re-lighting thing with a few cigars, just to see for myself. I figure if I don't have time to smoke a cigar completely, there's no harm in letting the stub go out and then trying a relight later. I mean, I've already thrown it away once, right? At the worst, I'll just end up throwing it away again.
It's been my experience that some cigars relight just fine; the stale-cigar flavor will be almost gone by the time you finish getting it relit, and completely gone very shortly thereafter. If you've got two inches or more left, there's a significant amount of enjoyment that remains to be extracted.
On the other hand, some cigars don't relight well: they continue to taste stale as long as I have continued to smoke them (although my spirit of experimentation has so far not led me to smoke any such cigar clear to the nub just to make sure the horrible taste doesn't go away).
Providentially, it has so far happened that it's been the cigars I really like that have relit well, and cigars that I didn't really care about that didn't.
I'm not sure what the standard accepted government-approved method of relighting is, but here's what I do.
First, blow a gentle puff backwards through the cigar to clear out the trapped smoke, so that when it goes stale it does so somewhere other than in your cigar. (If you don't do this, you won't have a good relight experience, I promise.)
After you do this, make sure no more smoke rises into the unburned tobacco on its own--hold the cigar either level or with the burning end up.
Now put the cigar down somewhere--preferably outside, because it's going to start to stink--where it won't get into any trouble. Tilt the burning end up a little, if you can. Usually when I end up with a cigar I can't finish smoking, it's because I've just arrived at my destination in my car; and there's a little groove between my left-side external mirror and the roof pillar that holds my cigar in the perfect attitude. (It works especially well if it's not raining, of course.) Do not put the half-burned cigar anywhere near unburned cigars, and don't put it in a carrying tube or case that you later plan to use for unburned cigars.
When you get back to the cigar, it ought to be cold and stinky. Tap it energetically but not destructively to get all the gray ash off the end; you ought to be left with a concave black surface. (Sometimes I use a tiny twig or a leaf stem to gently scrape off the last vestiges of gray.)
Blow through it again. (This part is probably pure superstition, but I do it anyway.)
Okay, here comes the unpleasant part. Put the cigar in your mouth, flick your lighter, and screw your face into the proper anticipatory expression of distaste. Now apply the flame to the foot of the cigar in the same pattern you used to originally light it (no need to toast the foot) while executing a quick series of short, hard puffs. Make sure you relight the entire foot; it's a good sign when you start getting a big yellow candle-like flame as you draw. Since this light is a bit closer than you're accustomed to, be sure not to singe your eyebrows.
Take the cigar out of your mouth; close your mouth; inhale through your nose; and exhale through your mouth, purging all the bad relight smoke. While you do this, just for a lark, turn the cigar around and hold the freshly-relit foot in the stream of exhaled air. Isn't that pretty? An ashless, relit cigar foot is much prettier than what you get on the first light.
Now try the cigar. Hmm. Not that bad, actually, is it? It picks up pretty much right where you left off; you don't have to go through the first few inches to get there again.
The best cigar I've relit, so far, was a Don Tomas (Honduran) Maduro in what they call Corona size, which anyone else would call Robusto. I mourned having to put it out early, because I was really enjoying it; when I relit it, all the enjoyment was still there, and I got another good two inches out of it. Mmm.
