I just got to looking at that cover more closely, that appears to be a 320 cover, so i went and and looked at my 320 turn signal switch and I suspect that you have a 320 turn signal assembly also.
If I were to guess I suspect if you look at your first photo you can see them 4 buttons to the left of your fingers holding the piece your asking about, so the piece you have in your fingers rides on them 4 buttons, it goes on the assembly you removed on a brown non-metal piece, it should make contact with them 4 buttons to make the turn signals work.
OK, so I just pulled mine apart to see how that piece between your fingers fit in the assembly, I regretted it afterwards as it was a bitch to get back together.
Is this what your assembly looks like?
Below is a photo of it before it was pulled apart, right in the center of the photo you can see the ends of your piece in the slots.
In this photo you can see your piece and the spring that goes under it to hold it against the buttons I referred to.
In this photo I put the piece where it belongs without the spring in place so you can see where it goes.
So what I had to do to get it back together was hard to do, did you see 2 little ball bearings fall out also, the two posts sticking up have springs in them and the ball bearings sit on the ends of them springs in the posts, so I set them on the posts holding that piece upright, that piece between your fingers I squeezed just a little closed, I put the little spring in place then I forced the copper piece in the slots and it stayed there because I squeezed it closed a little before inserting it in the slots, now I dropped the assembly over the piece with the ball bearings sitting on the ends of the posts and I made sure that part fell in and held the bearings in place, then I picked it up and started pushing the little arms in and over the assembly that returns the arm, what a bitch that is, once I got it in there I held it together and tested the assembly to make sure it worked properly, once I figured it was working properly I put the spring clip on it that holds it together, once everything was working for sure I took a small straight edge screw driver and carefully stuck it between the end of the piece and the block it sits in and spread it just a little till I knew it was loose and would ride on the nubs.
None of this was easy to do, if you didn't find the spring and 2 small bearings then you have a problem, as the spring holds the copper piece between your fingers against the nubs, and the small bearings make the assembly operate smoothly.