[JonChesterfield]

Relatively successful was slotting the cat5 jacket, cutting off two or three of the pairs, twisting/tying the remaining pair to the old wire, then sliding the jacket back over the join before wrapping in a conservative amount of electrical tape. You want the join to be similar width to the cable and preferably flexible.

I have a suspicion that pulling fishing line first is the right play if you can manage to connect it to the old wire. Flexible, very high tensile strength, small.

[retired]

In addition, in one room I ran two CAT5E cables as there was conduit along the entire way. So I took a CAT5E cable double the length of the conduit, stripped the outer sheath in the middle, folded the cable to get a loop and then attached the phone cable to that using the individual inner wires. Plus tape.