Verizon did it again. It's 10am in Tucson and the phone says 11am. Undoing "automatic time zone" and selecting the right time zone on the S3 works, but as the OP said, leaving it that way means it won't adjust when traveling.
Rebooting the phone works since it forces it to resynchronize and it appears Verizon has fixed the issue in the network.
I understand your concern, I know it's important to have the phone display the correct time. The phone is functioning as designed; in this case, the phone can only follow our network or you can manually change the time and adjust it. Our network complies with daylight savings time, so if the time is set to adjust with us, it will reflect the time as the rest of the country. I'm sorry,Arizona is does not compy with the time change. You can change when you travel, so that it can update.
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With the start of DST today, our phones were both off. Saying that the Verizon network recognizes DST means nothing. Your are able to set the local timezone appropriately by knowing where the phone is located; this is the whole reason for this setting. I have rebooted the phones and they now seem to have the correct time, but we'll see if they stay in this mode.
This has happened again this year; the Verizon network forced our phones to change. I can only assume that all Verizon customers in Arizona have to deal with this twice a year if they have the automatic time zone option on. That option should follow the local timezone, which in this case it does not. Verizon is just being lazy in not setting their Arizona equipment to MST with no daylight savings. All modern OSes have the Arizona timezone in them. Fix it.
Was speaking with an AT&T friend today that lives in Arizona and apparently the same thing happened to them so it appears that this isn't a carrier specific issue. They were able to get it correct simply by rebooting their device which they said they were already aware they would need to do.