I stayed in private rooms almost exclusively from Leon to Santiago, and suspect I could have easily done the same from SJPP to Leon had I chosen to. In his guidebook,
Brierley does list some non-albergue options, but I never needed any list or guidebook for finding private rooms. I never reserved a private room once, and never had trouble finding one upon arrival. The only exception would be the very small, remote stops where the albergue is the only bed in town. And this was during peak season (August).
The growing popularity of the CF seems to be resulting in more competition, so a rise in more sleeping options. I can't speak much to this cuz I've only walked it once, this past August. But I suspect the options were fare fewer 20 years ago.
A very useful term to memorize if seeking private rooms: habitacione individual? In most towns I often had the luxury of checking with two or three proprietors before making a choice. I also stayed in some very nice 3-star hotels for €35, sometimes less.
If you absolutely need private rooms every night of your Camino, some sort of accommodation list might be needed, for peace of mind if nothing else. But if you have a bit more flexibility, just showing up in a city/town/hamlet along the CF and finding a private room should not be a problem.
Hope this helps.