Just for the record, I have an AT&T iPhone 5, iOS 6.1.2 on StraightTalk with a brand new LTE compatible SIM card, running on AT&T's network. This does not make any difference, what kind of phone or carrier you use. All combinations have been reported to have this problem. Apple, AT&T, Verizon, Sprint...all of them deny to acknowledge the problem and fix it. Is it the phone, is it the network, is it both? We as consumers don't know, but they DO, YES THEY DO. They just won't acknowledge it.
I've read this whole thread and MANY alike. I found a workaround of the problem. It won't fix it per se, but it switches to LTE every time you do this.
The problem I have is that the phone defaults to 4G instead of LTE, and this does not bring only Internet speed issues, but drains down battery like crazy. Literally, if I leave my phone in this state, battery will go from 100% to 5% in a few hours only, without touching the phone (let alone using it). If I get it to connect to LTE, I get awesome battery life. I am not an expert, but I explain this with iPhone's constant search for LTE signal and when not able to switch and stick to it, it gets stuck with 4G. This loops forever causing the battery to go dead very quickly even if fully charged.
Those with the same problem already know that carrier WILL NOT help, flipping Airplane/LTE/Data switches or rebooting does not help either. Nor will reset network settings do any good. Replacing SIM card does NOT help. Replacing the phone with a new one does NOT help...
Here's what does.
While the phone is trying to connect to LTE and can't, it falls in a deadly loop which seems endless. What happens is that phone will have a normal signal strength with 4G, then all of a sudden signal drops to 1 bar and data connection is completely lost. This will last about 5 seconds. Then everything is lost, phone displays No Signal for about a minute or so and then reconnects with normal signal strength and 4G data. This loop won't happen hourly or daily, it happens every 5-6 minutes...you can't normally use your phone, if anyone calls you it would go straight to voicemail. It's THAT ridiculous. The trick is to catch that 5 seconds time frame when the loop begins, the phone will drop off the data connection and signal will go down to 1 bar, these 2 events happen at the same time. You need to make a phone call right at this moment. If you do, you will notice that signal strength will be restored to whatever it was before the loop started (say 3-4-5 bars) and in a second 4G connection will come live while in call. The call will go through, make sure you don't call a random number but you can call for example your voicemal. If you wait 10-15 seconds and hang up, LTE will come back, GUARANTEED.
Here it is summarized:
1. Lock your phone so you'll trigger that connection loss cycle.
2. (You may need to wait a few minutes, you have to try). Unlock it and catch it when it drops data and signal goes to 1 bar (these happen simultaneously).
3. Before it looses connection completely, quickly place a call (say voicemail).
4. Notice how signal bar goes back immediately to whatever it was (3-4-5 bars) and 4G data connection is restored. Loop cycle is broken at this point and the call will go through.
5. Keep the call on for about 10-15 seconds and close it.
6. LTE will come back shortly. Can't tell how long your phone will keep LTE on, but if it drops it for 4G, then repeat.
Hope this helps. Apple or your carrier will not help. Only thing they offer is flipping above mentioned switches, upgrading iOS, resetting network settings, exchanging SIM cards or phone itself. None of these will help, tried it.
Cheers!