That hose looks like it might be the lower-lower hose.
I replaced mine recently and I remember I had to trim the upper-lower hose where it connects to the T fitting. I remember the hose was super tight going on the T valve and was a real biyatch to get it to fit. Had to grease it up and really work it on.
The lower-lower hose gets skinnier where it connects to the radiator. I got my hoses from CCOT.
Interesting. The way I saw it the lower hose fit great, big at both ends, but the upper one was the one with the taper. That is the one that I could not envision fitting. Can someone with an OEM hose chime in? Another case of the non-OEM blues?
I just replaced the hoses in the pig. The large diameter hose should go to the water pump and the smaller diameter will go to the t-connector and another hose from the t-connector to the radiator. I just take my old hoses to Napa and get them there. It might take some cutting to get the right length or form, but it works out. You want to make sure you get hoses that fit pretty snug and not too loose, so they'll clamp down nice and tight.