Well depending on the phone you put it in that could be an issue.
Also, the contacts would not be on the SIM. They would be on either the internal storage of the phone (that has the broken screen) or any sd card that was/is in it. It would depend on if they had been saved to the storage or the card. The same goes for any other data like photos, videos, music, etc.
Any text messages you had on the phone that is now broken would not be accessible anywhere else. So unless you can get them off the old phone (the same goes for the other info) then they are gone for good.