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.

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.