It's possible that your text messages were stored in your old phone's memory rather than on the SIM. You could try replacing your old SIM in your old phone and checking where your texts were stored. You may find that you are able to transfer them from your phone's memory to your old SIM, then across to the new phone with the SIM as you have done with your contacts. If that doesn't work, there are SIM readers that you would have to buy together with the software to run on a PC see - www.justaskgemalto.com/us/what-uicc-and-how-it-different-sim-card/ . It seems a bit like overkill though. I'm not sure if your phone provider can help with this, although I doubt it but you could try calling them. Also, older phones didn't store very many messages anyway and older messages were just deleted. You may have to open your old phone and just check through your texts making a note of important information the old-fashioned way.
When you set up your new phone, you could use a SMS backup type application on it. I have this set up on my Android phone and all my SMS texts are backed up to a folder in gmail - it's worth considering.