Might depend on how the old phone stored data. Possibly on the SIM itself?
Might be possible to use a SIM card reader?
Did this myself years ago, with an old Nokia. bought this cheap SIM reader, and popped the Sim into it, and read all the data from it, then swapped for the new sim, and copied to it?? Got my contacts and some texts ported across..
Bought reader from E-bay for a couple of quid? All depends on how old the phone, whether it used phone memory of memory card etc, but might be worth a try??
(even if the old sim is no longer active, anything stored on it is still there, it just can't access the phone network)