I was wrestling with a dry cork and it snapped in half. I eventually had to push the jammed piece down into the bottle to liberate my wine. Embarrassing! Is there a better way?
Don't be ashamed. It happens to the best of us. And, yes, there are two better ways, assuming there's enough cork left to play with inside that neck. My first plan of attack is always to pierce the cork with the tip of the corkscrew while holding the corkscrew at an angle (versus going straight down as normal). You'll be frustrated and impatient, which is understandable, but it's critical to go slowly. As you pierce the cork, apply as much sideways pressure against the bottleneck as you can while turning the corkscrew. This will help maintain friction between the cork and glass and – hopefully – prevent the piece from sliding down into the wine. Once you've inserted the corkscrew all the way through what's left of your cork, gently pull upwards. Remember: I said "gently."
For this technique, it's best to rely on a basic old-school corkscrew with an unencumbered helix, such as a "waiter's friend" (sometimes called a sommelier's corkscrew), versus the two-armed butterfly models or a fancy Screwpull type.
The other method is to use a special device known as a two-pronged cork puller or Ah-So, which you can find at most good kitchenware stores. It's tricky to use, and you might want to practise first on an undamaged cork to get the hang of it. One prong of the puller is longer than the other, and you want first to insert the tip of the longer prong down between the cork and bottleneck, being careful not to push the cork down any farther. Once it's between the cork and neck (say, two or three millimetres deep), insert the shorter prong on the opposite side and rock the handle back and forth while applying gentle downward pressure (first with one prong, then the other) so that both prongs extend beyond the bottom of the cork. Then turn the handle while pulling upward.
Not all broken corks will co-operate, so don't blame yourself if things go awry. The worst that can happen is that you'll end up with a broken cork bobbing around in that bottle like it did the last time. The only guarantee against such minor mishaps is to drink screw-cap wines.
The Flavour Principle by Lucy Waverman and Beppi Crosariol (HarperCollins) won top prize for best general English cookbook at the 2014 Taste Canada Food Writing Awards.