...get what the people in the trade call a snake. If you go to rental place get a motorized one then you don't have to do as much hand work. This is usually a tubular device which you shove down the drain while in insert turns to clear any debris.
If the blockage is further down the trap, as you say, then the snake will get to it and move it along or at least allow you to snag it and pull it back out. Make sure to get the right diameter snake head for the pipe.
If the rental place has one get a snake with a camera head end on it. You can then see what down in the pipe.
The other plumbers' trick they use is after the blockage is cleared they ask you for three or four pots of hot, boiling water to pour down the drain. This also helps moves down any debris clinging to the sidewall of the pipe.
If you are already working at this think about replacing your trap/gooseneck with one that has a removeable plug at the bottom. This allows you to easily do regular clean up maintenance of the uglies that settle to the bottom of a trap and do not decompose in water. Things like human hair, toothbrushes, dental floss, etc.
Usually a blockage occurs because an object that doesn't decompose gets lodge in the pipe and then other nasties like mucus, etc clings to that and builds up eventually blocking the drain.