Is it possible to do a renegotiation of the SSL handshake while the current SSL connection remains active. When the new handshake has succeeded the server should respond with a confirmation of the new handshake.
I've searched on SSL renegotiation but I couldn't find anything concrete. Does anyone know if something like this is possible?
Yes, renegotiation is part of TLS protocol. It can be both client- and server-initiated. Support for it depends on the implementation. Also, renegotiation made an attack on TLS possible.