Seems to me you already have all the parts you need and won't have to buy anything extra.
I've been drinking ro water since I put in the system for the aquarium. Still this side of the grass!
This link will help you to "see" more clearly Shao
http://www.watergeneral.com/support/pdf/RD102.pdf
Refer to page 9 rt hand diagram, model RD106
Not affiliated, just a diagram I found on the net.
The 5th stage would be your DI, the 6th stage is just another carbon filter to get rid of the "rubber" taste of the storage tank.
Where it says to RO faucet, thats the faucet you installed on your counter, close to the sink, so you could have drinking water, before it gets to your fridge. If you have it this is where you would tee off to go to your fridge.
If you don't have that faucet, then go straight to your fridge outlet from there.
Hope that helps