You have my sympathy! With three girls in the family, we always have hair balls plugging the drain. Additionally, I own many rental houses with tenets who don't understand that forks jam a garbage disposal and cooking grease plugs drains.
I usually just remove the entire "P" trap", then run the snake in from the nearest clean out, usually located on the outside wall or close underground near the sink (or whatever needs cleaning)
BTW, that's an old timer, I haven't seen a trap like that in years, the're now all plastic.
I forgot to mention, removing the "P" trap and flushing it out with a garden hose usually does the trick. The trap's function is to provide a barrier to keep sewer gas from entering the house, but it also catches most of the debris and is usually the first thing to clog because of it's tight bend. Kitchen grease is the exception. Additionally, a garden hose can sometimes be used instead of a sewer snake. Jam the hose as far up the clean out as possible and turn it on while moving the hose back and forth.