I recommend Barge Cement:
Barge Rubber Cement
It is a waterproof and flexible contact cement, better than Shoo Goo in my opinion for this purpose. I was a wading boot tester for Weinbrenner Boot Company, the makers of the Gary Borger Wading Boot. This is what they use in the original manufacture of the boot before also sewing the felt on the boot welt.
The reason I think Barge Cement is better is that it will from a thinner layer and hold just as well, plus it is a contact cement.
Coat both surfaces and allow to get tacky. Once the surfaces are mated, they will not move so line them up carefully.
The best way to apply compression is to cut rubber strips from an old tire inner tube from a truck tire. Then wrap these as thick rubber band around the felt and boot to apply compression. Wrap as tight as you can. This will give you an even compression between the new sole and boot bottom.