Is there a secret sharing scheme which allows delegation/re-sharing without reconstructing the original secret? EDIT: Ilmari Karonen's answer below well not exactly what I want, gives a very good idea of what I am trying to accomplish. 

Are there any known secret sharing schemes that allow new parties to be read in on a portion secret, preferably without all parties having to be online at the same time?
I don't want to have to keep the original trusted authority around. The question is, is this possible without reconstructing the original secret?

My idea is that I divide a secret between $N$ of $k$ people at time $T$. At time $T+1$, I want to change it to $n$ of $k&#39;$ people. This may or may not be strictly possible, but is there something possible along those lines?
Suppose we have a $(k, n)$ threshold scheme meaning that there are $n$ shares of a secret distributed to different parties, and any $k$ shares can be used to re-create the secret.  A new person joins the club and wants to have a share of the secret too.  I contend that the secret must be available to a trusted party who can create the extra share.  Because if someone with little knowledge of the secret (or even as much
knowledge as a cabal of $k-1$ shareholders trying to break the scheme) could
create a new share of the secret, then this someone could repeat the process many times until $k$ shares are available, and thus recover the secret via the standard reconstruction
technique.  Now, it is possible to have a mathematical algorithm or computer program 
that will take $k$ (or more shares) and create new shares without the secret being
explicitly reconstructed, that is, none of the quantities used internally or stored
anywhere (register or memory cell or disk or tape drive) will actually be the secret itself and so the new share creation process is safe from the casual eavesdropper.
However, a savvy opponent  who can view the process or get a core dump from the
processor  will be able to recreate the secret.