Unclogging bathtub with an external tower drain
I think what happened is, when the snake got to the "T" where the horizontal pipe meets the tower, it went UP.  

A metal snake is always bent a little, which makes it tend to go one way or the other when it gets to a junction.  If you remove the tape, turn it over, and reinsert it, it might go DOWN instead when it gets to the "T" - the bend in the tape will make the tip point down instead of up.  

But it would be easier to just snake straight down the tower.  You have cleared the pip from the drain to the T, so going straight down the tower should be fine.  

Just remember to go nice and easy with the snake!