SIM card swapping is one of the best idea to avoid huge bills on roaming. If the new phone is free from network lock, Then your SIM card must work fine with it. But make ure the phone was unlocked safe using unlock code. Certain unsafe unlocking methods may give problem in working with other networks.
So it is better to unlock the device yourself using unlock code. You can get the unlock code from any private service online. You can also visit
http://www.mobileunlockguide.com/ to get the free unlocking instructions. Good luck.