The problem was that if I tried to add Photoshop Elements 10 as a supplemental editing application in the Elements Organizer 11 preferences, I got a message that the 11 organizer already supports Photoshop Elements, and my PSE10 selection was rejected.
I found a way to launch Photoshop Elements 10 from the Elements Organizer 11 menus and buttons by creating a copy of PhotoshopElementsEditor.exe in the root of the programs folder for PSE10 and renaming the copy to PhotoshopElementsEditor2.exe. Both PhotoshopElementsEditor.exe and PhotoshopElementsEditor2.exe are in the same folder. Now I can go into the Elements 11 Organizer preferences and select Photoshop Elements 10 (PhotoshopElementsEditor2.exe) as a supplemental editing application. This allows me to select a photo in the Elements 11 Organizer and edit it directly in Photoshop Elements 10. I think I forgot to mention that I don't have Photoshop Elements 11.
If you have done something similar but later have run into problems, please let me know. So far I haven't had any problems.