I've just done a tuning job on a Lightning XL which has an identical trigger assembly to your Supersport.
For purposes of identification of parts I'll refer to the part numbers as quoted on this diagram: http://www.gunspares.co.uk/showcatimage.asp?id=24314
borrowed from Chambers site!
Firstly I don't think you needed to order any new trigger parts as those in your photos look in good serviceable condition. Shame but now you have some spares just in case of future problems.
Secondly, you can't really put the trigger together without the piston has been refitted and it's far easier to fit the spring and spring guide without the sear in the way.
When your piston, spring, guide, cocking linkage and crosspin (DP108) are refitted ( I assume you have a spring compressor or a patient trustworthy mate who weighs enough) refit the sear (SR028) and the largest diameter parallel pin DP027. The "waisted" pin is DP047, which is fitted later.
Fit CS164 the sear spring as you have fitted it in the above photo using one of the short dowel pins DP028 (there are 2 identical length and diameter) through the front holes of the trigger housing.
Next assemble the trigger blade and trigger sear (TR033 and TR032) and fit these between the sides of the trigger housing, retain them with another DP028 through the lower pair of holes in the trigger housing.
The spring CS163 is now fitted, the "crossbar" engages with the trigger sear in the square corner where the trigger adjusting screw (C1055) pokes through, it's best to wind it out a bit, until the screw is no longer visible in the square inside corner of TR032. The knee shaped angle of the spring points upwards as the rifle is lying on it's back on the bench.
Take the longer of the small diameter dowel pins (DP045) and thread it through one side of the holes behind the holes with the first DP208 through. Catch the loop of that side of the CS163 spring on the pin, push the pin just far enough to almost catch the "tongue" of CS164. Now you need to press down with a small screwdriver on the tongue until the pin can be pushed across it, then align the other loop of CS163 and push the pin through. The pin is "too long" but the excess should be left protruding at the safety catch side or the stock won't fit.
Now the trigger sear needs to be pushed slightly forward and the waisted pin DP047 fitted behind it to keep the sear from falling out of engagement.
Simple if you have 4 hands!
Apply a few drops of oil and check the safety catch moves, then refit the action in the stock and see if it cocks. It's not a good idea to try to cock the action out of the stock as there is a cocking linkage guide in the stock.
I think I've remembered it all in the right order, I'll try to answer any questions if you don't understand what I've said.
Don't swear at me when the springs fly across the workshop!