Secret sharing scheme with ability to add or update share number I would like to have a secret sharing scheme with the following properties:


Generate secret $S$ a split it to $N$ shares $(s_1, s_2, ..., s_N)$ 
Require $k$ of $N$ shares in order to reconstruct secret $S$
Update $N$ to $N+1$ and generate new share $s_{N+1}$ (everything remains
the same, $N$ shares not changed)


The scenario is that I use Shamir secret scheme to split secret $S$ into for example 4 parts and require 2 of them to be present in order to reconstruct $S$. In that case if I would like to reconstruct S then I have to also input scheme type, i.e. 2 of 4.

What if I need to update number of parts but need to preserve existing shares and be able to reconstruct the same secret $S$? Is there a way how to do it?
With Shamir's Secret Sharing you can add as many shares as you want, as long as the threshold and the secret is unchanged. 

You can see that neither the generation of the polynomial that produces the shares nor the reconstruction of the secret by polynomial interpolation require the parameter $N$ (number of shares), but only the threshold $k$. 

Of course the number of shares $N$ should obey $N\leq p-1,$ since they need to be obtained as polynomial values at distinct points of the field $F_p$ and one point, usually zero, is reserved for the secret. You can increase $p$ if needed, but shares need to then be recomputed modulo the new prime you choose.