Relighting a cigar can be a tricky business. One technique is to clear the entire foot of all ash by gently rubbing and/or tapping it on the ashtray. You then re-toast the cigar to get the wrapper going. Relight the cigar by puffing, then blowing OUT through the cigar, which is supposed to clear the foot of tar and resins left from the previous light. The downside of this technique is that sometimes you blow out sparks, turning your cigar into a Roman candle.
I’ve tried this in the past with some success, but I stumbled upon what I think is a better way to go. Cut the foot of the cigar about 1/4 to 1/2-inch back. This leaves you with mostly “fresh” tobacco. However, you may often find the tobacco in the center a little scorched or blackened depending on how deep you cut it back, but it will help keep the cigar from tasting bitter.