You need to try to get the cover off of that old drum trap and clean it out.
But it might not budge, or you might bust your pipes while twisting on it.
Penetrating oil applied several times over a few days might help.
And gives lots of small taps to the edge of the cover, to help the oil penetrate.
You will often have to just drill a series of holes through the drum trap cover and break it out of there.
Big box stores and plumbing supply houses sell new covers.
Meanwhile, how long was your snake?
A clog can easily be 30 feet away.
I have found some almost 50 feet away. Especially in a ranch house.
And a snake can poke right through a hair and soap clog, and when you pull the snake out the clog just closes up again.
Proper snaking involves lots of rotation, and moving forward and back, to break up clogs, not just to drill through them.