Can you put the t-case in neutral and flat tow it?
BTW, pulling your PTO cover isn't going to show you anything useful, your only bet is to unjam the shifter somehow, and pop the top off.
Most popping out of 3rd problems are shift fork related, not syncro as is the most popular suspect for some reason. Once you get the tranny apart, you'll see what I mean.
All the syncro does, as it's name suggests is syncronize the speed of the shift collar with the gear on the mainshaft. It does this by putting a soft friction material between the gear and the mainshaft, until they are the same speed, and allows you to pop the gear in, as if the transmission wasn't moving. When guys powershift / double shift, they are just manually matching the speed of the shift collar and the gear by revving the engine up, then the gear dumps straight in as if it was syncronized.
The syncros are made of a far softer material than the gears are, and about all that the syncros can do is wear out, and not spin up the gear you want, causing the shift collar to grind against the splines on the gear.
Another thing that some people claim causes 3rd gear popouts is a snap ring on the countershaft (bottom gears you see when you open a PTO cover) that weakens over time and allows the gear to shift slightly forwards. But, I personally don't see how, as it wouldn't make much difference if the bottom gear is walking around a touch. It won't make the top one that has the shift collar slid onto it pop the shift collar off. I put the little sleeve that they sell aftermarket on not to prevent gear popouts, but to prevent a pretty catastrophic failure if that snap ring were to actually break, and allow the gear to slide forwards on the mainshaft, lol
However, if your shift forks are worn out (they have a plastic coating on them from the factory that wears off), or if you've got a lot of slop in the shifter then, its gonna cause the shift collar to not engage all the way onto the gear, allowing it to slide off without much effort at all.
