I just replaced my serpentine belt and did this without removing the fan as several threads on this topic have suggested. It worked really well and took me about 20mins. Just wanted to share the steps that seemed to work well for me in case it helps someone. Since I am doing an oil cooler, and turbo replacement I had the alternator removed while doing this.
First step is to draw a quick diagram of belt routing with old belt in place, then remove belt. It took me actually more time to remove the old belt since it was so stiff, I ended up cutting it and pulled it right out, should have done that from the start
From passenger side feed one end of the belt, keeping it above the tensioner, to the big pulley at the bottom front of the engine under the fan.
From below engine, driver side, pull the bottom of the belt loop over the large pulley. That part of the belt can now also be routed over the pulley next to large pulley and then under the pulley next to that one at side of engine front.
From above engine passenger side route the bottom side of the other end of belt under the pulley below the tensioner.
Then fold that part of the belt around the top of the tensioner pulley, it is a little trickey but there is enough room to do it.
After reinstalling alternator I can just move the tensioner (with tensioner release bar ~$10 from parts store) and loop the belt over the alternator pulley.
Hope this is helpful.