When I purchased 5 in 1 reflectors, I got the kits that include a stand. Westcott make this piece that fits on a light stand. http://www.henrys.com/62205-WESTCOTT-REFLECTOR-HOLDER.aspx The kits come with it and the stand.
What about a chair and a couple of spring clamps? Or, a chair and a book to keep it from sliding off the seat? I saw a video on lighting earlier today that included a reflector hung from a microphone stand using three spring clamps. I guess it depends how handy you are and what you have available.
Emf makes a good point — you should rotate just your eyes more toward the camera, once you have your head in position. On the other hand, being on the right third line is good, with eyes at about the upper third line.