Even if his phone was connected only to WiFi and in airplane mode, the time should change as the plane moves and connects to different towers.

And there are tons of internet-based messaging apps he could have been using. iMessage, as noted, is one. Hangouts, which you can integrate with SMS (or has SMS integrated automatically on some devices). Plus who says it was SMS - unless you specifically looked and saw. In which case, was it iPhone or Android?