Hmm. So the new phone cannot detect the old SIM card. Can you put the old SIM card into the old phone (presumably it is detected there)? Can you then copy the data from the old SIM card to the old phone?
Then put the new SIM card in the old phone and copy from the old phone to the new SIM card - Bob's your uncle!
Another option is to use a computer. You can buy a SIM card reader to plug into your computer. Copy from the old SIM card to the computer and then from the computer to the new SIM card.
Some phones have Bluetooth. Your new one probably does. This means you can transfer information without any wires. If your old phone has Bluetooth you can transfer data this way (although some phones only let you transfer one phone number at a time, which would obviously be extremely tedious).