Almost every time I plug in my phone to recharge, it will then tell me "no sim" and I have to turn it off and back on for the SIM to be detected. I never had this problem with my 4, just now that I have the 5S. Any ideas?
Is there a chance that your SIM card is not fitted in the SIM tray properly or that the tray is not pressed down into the phone all the way? IIRC, is should make a "click" sound when it's fully inserted into the phone.
If you have had the phone for less than 14 days from the original date of purchase, I would consider returning or exchanging it. You may need to make an appointment with Apple to have the phone looked at and tested in one of their store locations.
As a last resort, you might want to consider getting a new SIM card from a corporate AT&T store.