Dublins and conical bowls are my favorite pipes, but it's very rare that I smoke it down to the last grains of tobacco. It happens, but it's not a big deal. We have this whole word for that last little wad of tobacco that is left over, "dottle," and if it were just something that occurred when something goes wrong, we wouldn't have bothered giving it a word of it's own. Dottle happens. Embrace the dottle.
Sucking flame down to that last little bit, or puffing harder to smoke the dottle is most likely going to lead to burning up some of the delicate wood around the draft. There are no rewards for smoking it all down to ash.