Contact your current provider and ask them for a PAC code. This is valid for 30 days. Once you have your new sim, activate is as normal. Wait for it to be fully active (up to 24 hours). You will be given a new random number.
Then go to this page:https://giffgaff.com/profile/details#portin
Enter your PAC code and your number will be ported across.
It can take up to 24 hours to fully come across though, so keep restarting your phone, and you may find some services intermittently working in those 24 hours.