The unburned shaft of the cigar filters out the flavors from the burning tip. As the cigar burns down the filtration becomes more and more saturated allowing more flavor to come through. Also, the filtered out deposits burn again and are released with increasing effect. There is a point at about the last 1/4 of it's length where the cigar will change very quickly and, most would say, become objectionably stronger. I've often thought of proving this with a mathematical model but have not yet done so. There are some connoisseurs who toss the cigar at that point. I assure you that I am not in that class but I do find that with a 7-inch stogie the first inch is rather wasted.
I've never had to wait for half of it to be gone, though. It sounds like you would benefit from a stronger smoke. Try a Punch or Joya De Nicaragua Antano.