Are you sure that your contacts are stored on the SIM?
Very few people/phones store contacts that way these days.
Normally they're stored as Google contacts (Android) or iCloud contacts (iPhone) and backed up to the cloud automatically and can be restored from there too.
If they are stored on the SIM and you are on Android or iPhone then you can convert them to the relevant cloud based contacts easily.
(I'm sure that there are lots of other guides on how to do this on the web).
What phone are you moving from/to?
Hope this helps.