Is there a secret sharing scheme which allows delegation/re-sharing without reconstructing the original secret? In the typical programming scenario, it wouldn't be created then evaluated. It would just be evaluated at xi by summing the values of the basis polynomials evaluated at xi. The same programming construct the parties use to evaluate L(0) can be used to evaluate L(xi). Not sure if you need me to recreate the steps or the polynomial here, or if you are getting at something else?