iPhone 6 — "No SIM card" message

I have a 2-day old iPhone 6 that started displaying a "No SIM card" message about half an hour ago, out of the blue. It was working fine one minute, and the next minute it wasn't. I tried rebooting the phone but that didn't work, and nothing I have on hand is small enough to pop the SIM card tray open. Does anyone have suggestions for how I can fix this myself, or do I need to take the phone to the Apple store? Thanks in advance!