Have you tried to reboot your device? on/off + home button. hold down till you see Apple logo & let go. (ignore the "slide to power off")
My SiL updated her iPhone4 to iOS6. She lost her contacts "EDIT" function (top right corner of screen on a contact) as well as her contact search from the phone app. I told her only way to rectify the issue was to do a restore thru the comp/iTunes.
I know, you don't want to hear it, BUT it may be the only option that you'll have to do a restore again. A member here, i recall when iOS6 was released, had upgraded. Had an issue - cant remember exactly what, but she had to restore twice to have the iOS load properly & correct the issue she was having.