Apple doesn't store contacts on a SIM. Sign into iCloud on your other device and as long you had your old device set to sync Contacts to iCloud they should reappear on the new device. If you did not then you will need access to a computer backup.
Click on this link to learn how to 'Set up iCloud on your iPhone, iPad, or iPod touch' --> https://support.apple.com/HT208681
See step 3 on how to select specific items to sync.
Click on this link to discover how to 'Restore your iPhone, iPad, or iPod touch from a backup' --> https://support.apple.com/HT204184
or watch this video if you're using a Mac to do this:
https://www.youtube.com/watch?v=Jbp3gGTQygg