I'd almost be willing to bet the issue is with the battery itself, particularly if it is the original battery supplied with the camera. The Rebel T3i/600D was introduced in 2011, but the LP-E8 battery that powers it debuted earlier with the Rebel T2i/550D in 2010.

No rechargeable battery lasts forever. The LP-E6 battery I have that was made in 2011 is no longer usable. I have a couple of newer ones made around 2014 that are also on their last legs.

The big question, though, is whether the battery damaged the camera when it discharged its energy fast enough to get hot like that. If you've got a friend with a Canon camera that uses the LP-E8, borrow one and see how your camera does with a different battery. (The Rebel T4i/650D and Rebel T5i/700D also use LP-E8 batteries, along with the Rebel T2i/550D and Rebel T3i/600D)

Interestingly enough, in my experience, reputable brand third party LP-E6 batteries last slightly longer than OEM Canon LP-E6 batteries do. I use Maximal Power (now discontinued), Watson (from B&H), and SterlingTek. There are a lot of other third party batteries that are junk, so be discriminating in which third party batteries you use. I would expect LP-E8 batteries and their third party clones to be similar.

Canon batteries seem to demonstrate a noticeable gradual decline over several months or even a little over a year before reaching the point of non-usability. The third party batteries seem to do well for longer, and then just fall off a cliff in only a couple of recharge cycles at the end.