Here are the components of the seat. There's the pleated center section, the two pieces on each side of the center, then a band of royal blue going around the sides. The back of the seat is in the foreground.
The first time around, I sewed the center section to the left and right top pieces, then sewed the side to that assembly. When I got to the front, I realized that I had to use the seam ripper so I could get the side welting under the welting attached to the center pleated section. After ripping, tucking, and resewing, it looked great.
But I thought I just had the order wrong, so on the next seat, I sewed two the side to the left and right seat uppers, then sewed the left/right seat uppers to the center section. This is where I had to hand stitch. The first procedure yielded better results, but both look fine.
I think if you actually rehearse the sewing, instead of staring at it, coming up with a gameplan in the depths of your brain, then hitting the machine, you would arrive at the most logical way to sew it up.