I do exactly as hamiltont has described with fishing line and rag/bits of nylon scrubbie/whatever.
However, steel wool is a no-no with stainless steel (can cause rust).
I recommend you tie the rag in the middle of the fishing line line, so that you can pull it through the dip tube both ways or reverse it if you get stuck near the bend in the tube.
If, for whatever reason, the line breaks and the rag becomes lodged, I've found the wire coat hangers (bent into a straight wire) are very good for dip tube disimpaction. Tape the cut tips of the wire to protect yourself and the dip tube lumen from scratches.