I have never had a stiffness problem with basting spray, and I've used several different brands. My favorite brand, though, is 505. I think the June Tailor brand is the one that gets some complaints on this board. Could you possibly have used too much spray on it? It doesn't take a lot of spray (at least for the cotton battings that I use).
It's a good idea to secure the backing before starting. For a small quilt, I use painter's tape to tape the backing to my kitchen island. For a large quilt, I secure the backing (making sure it is flat and straight) to a large table with clamps.
Next step is to layer the batting on top of the backing. Once I am satisfied with the positioning, I peel back half the batting, spray it, then smooth it back down on the backing. Lift the other half, spray it, and smooth back down. I smooth from the center out to the edges.
Last step is the top, using the same method of positioning first, then peeling back half to spray half at a time.
I also, especially on large quilts, mark the center of each side for the backing batting and top to help me with the initial positioning.
The only time I have had to re-smooth a lot is when I sprayed the whole piece before positioning. That taught me to position first, then peel back and spray only half at a time.