Yep. Eye-Fi Card is about it. See Page 230 of your manual.
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Wi-Fi SD cards are terrible. The leading manufacturer of them went out of business. I wonder why.
I do not know if the external Wi-Fi adapter works with the T3, but I do know that it costs more than the camera. You would be MUCH better off buying an updated camera with built-in Wi-Fi and Bluetooth. The Bluetooth makes the Wi-Fi setup a breeze.
However, understand this. A Wi-Fi enabled camera does not mean that it is fully networkable. All that it really means is that the camera can communicate wirelessly with other wireless devices using standard Wi-Fi frequencies and protocols.
I do not know what functionality you expect from having Wii-Fi, but be prepared to be disappointed. On all but the most expensive camera bodies, built-in Wi-Fi is somewhat limited, and SLOW.
A wi-fi card seems to be the only option, your first point would only enable a standard SD card to be wi-fi when inserted into an wifi adapter, not the camera. Once you remove the SD card you might as well use a computer that would allow you to share the contents via wi-fi or Bluetooth or direct.