IPhone – use the T-mobile contract free iPhone 5S with the AT&T number

Afaik Verizon can only unlock the phone completely and not block certain carriers.

Until the iPhone 5, there was no model that supports CDMA and GSM.

The reason your phone doesn't work on an AT&T network (or most international networks for that matter) is because AT&T doesn't run a CDMA network.

http://support.apple.com/kb/ht3939

Unlocking appears to be a pretty easy process. Just call them up or go online. Here's another article from Apple about it.

After the phone is unlocked, you should be home free to swap SIM cards with no jailbreaking required at all. I've had multiple unlocked smartphones (Blackberrys which previously had data contracts) in the past, which come down to just pulling the SIM, and moving it to a new device. Apple started with just AT&T, and at that point made it difficult to change carriers, but things have changed, and particularly the iPhones 4/4S do quite well at switching networks. The two year data contract can also be confusing, because this is required if the phone is to be purchased at a lower price. After that, there is no ties, to speak of with the carrier that originally sold you the phone.

After the contract ends (and in this case, it is unlocked), you can use the device as a iPod Touch (wifi) + phone. You could even get a prepaid SIM card (from AT&T) and use it without unlocking it.

*Apparently (thanks to @jsd), you don't even need to unlock your phone. Switching to ST should be pretty easy for you.