> Which gives you both less room (is requiring you to use a thinner implement) but also a greater risk of damaging the socket.

A flattened wood toothpick works great for this. Take a regular round pointy toothpick and flatten the end 5mm or so by squeezing it in a pair of pliers or some such.

I was ready to replace my phone until I learned this trick. I had already tried cleaning the port out, but obviously not well enough.