However, the RJ-11 is the connector that connects the phone to the wall. Usually the connector that connects the phone to the handset is even smaller and also has 4 wires. Do you know what the correct code for this connector is? I found a document that seems to call it 4P2C, but results for that don't seem to be accurate.
Careful! A phone handset only needs two wires, and many handset cables have only two wires in the cable, and/or two contacts in the plug.
4p2c means "4 position, 2 conductor": there are four parallel grooves in the plug, but only two have metal contacts. What you would want is 4p4c: same shape plug, but four conductors.
4p2c is often unofficially called RJ9; 4p4c is unofficially "RJ10" or "RJ22".
If all you need is three conductors, a 1/8" stereo headphone jack might work nicely too.