Epson WF-3640 paper jam message, spring and gear fell out

While I cannot find a copy of the SM readily available for this model, what I would do is look near the paper feed where the jam door is and look for anything obvious that takes a spring - look for a component which is not in place, that will be the quickest way to tell.

That said if you can post photos of the back I might be able to find it, otherwise if you want to buy the SM it's not hard to find, like at all - Epson manuals aren't as widespread on the free side, but many sites do leak them and sell them for $5-10 so it might be worth purchasing and seeing if you can find a reference to the spring. Worst case, the printer is BER in some way, and you didn't spend much. For the $50-100 printers, they're disposable if you can't figure it out in an hour or two, but the $200-250+ machines are at least worth spending $10 on a service manual to see if you can figure it out, given the price difference of the printer.