@Kieran to create another share you need to have the secret. The secret is a curvy line defined by K points. If you know any K points on the line, you can draw the line, if you know the shape of the line you can pick another random point on it. Yeah, so to update the line you do actually need to hold the secret. If you backup the secret and forget it locally, then you can't do that.

If you really wanted to not hold the secret locally, I think you could create future shares that are held locally, but encrypted to curve25519 keys that your friends hold. You can't decrypt them, you need your friends to do that... but you'd just have your friends hold them one at a time. Hmm, I think there is still more investigation to do here.