Serpentine belt installation

[JParsons]

I am attempting to replace the serpentine belt in my 2007 1.5L Yaris. The old belt slipped off easily enough, but I’ll be danged if I can stretch the new one on. I can get it over the crankshaft, AC, and waterpump pulleys, but the alternater pulley (smallest and most readily accessible of the four pulleys) seems too far for the belt to stretch.
I compared the new belt to the old and they appear to be the same length. I called the parts store and asked them to reconfirm the part number (Continental 4040465) and they say it’s correct.
I finally unbolted the alternater from its top bracket so I can swing it down far enough to get the belt on, but no way can I swing it back up to the top bracket now.
There is no tensioner pulley or any other way to gain slack besides moving the alternater, so what gives? How is this supposed to be done?