Breaking the cork off in a wine bottle's neck is very annoying—and also kind of a party foul. Luckily, it's such a common problem that a mini-industry of homemade remedies and gadgets has sprung up dedicated to efficiently getting those stuck corks out. Two tricks work particularly well—one homemade, and one that uses a gadget. Both keep cork fibers out of the precious wine.
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Push the stuck cork down into the bottle with a wooden spoon handle or other narrow poking device. Push gently and evenly to avoid breaking any more of the cork.
Place a coffee filter across the spout of a decanter. Push the filter down a couple of inches inside so it fits snugly.
Pour the wine through the filter into the decanter, slowly to avoid spilling and to get every piece of cork into the filter. If your grip holding the filter and the decanter is slipping, take a break. If the cork pieces in the coffee filter block the flow, stop and dump them into the compost and continue.
Consider keeping a two-pronged cork puller on hand for times like these. The best ones have Teflon blades; the material is thin and strong enough to grip the cork properly.
Gently insert no more than 1/2 inch of the longer blade of the cork puller between one side of the cork and the inside of the bottle. Gently insert no more than 1/2 inch of the shorter blade into the opposite side between the cork and the bottle.
Slowly rock the puller back and forth in a see-saw motion to drive the blades deeper along the sides of the cork. Go slowly to avoid pushing the cork into the bottle.
Stop pushing the blades when they're gripping the entire length of the cork, plus 1/2 inch, if possible.
Turn the puller as you pull the cork upward and out of the bottle. The blades grip the cork as it comes out. Again, go slowly to avoid pushing any cork pieces into the bottle—and to avoid spilling wine on yourself.