My two .36 Senecas are very early 2XX serial number guns that was supplied with a breech plug wrench. I have in the past, removed their breech plugs and measured the "breech chambers." They are the same diameter as the one on my .32 Cherokee; .285." I suppose it is possible that yours was made with a larger chamber or that someone enlarged it. Thompson Center did make a lot of small changes to these guns while under production. Unless the patch/jag is entering into the breech chamber with force, it should come back out without jamming. There are a lot of factors at play here and without inspecting every one of them we are all only offering our personal experience to help you with this problem. I strongly suspect your choice of patching material and its condition (wet/damp/dry) is the culprit. Your jag might actually be a bit too small with your current patching.
I set up ALL my muzzle loading wiping/cleaning equipment to be used with unbleached cotton muslin cloth. I do not use flannel, old bed-sheets, or in particular; old T-shirt material. Washed, unbleached cotton muslin is thin enough to slide past the fouling but (if properly proportioned to bore size) will bunch up on the upstroke and pull fouling from the bore. It is also very absorbent. It is used slightly but uniformly damp. Since you are new to the sport, you will find there are a lot of details you will have to work out for your particular gun to develop a system that works for you. Most of what advice you will get is how things work for us--YMMV.