You can try a bigger snake. Rent one. It's worth a shot, since it's the exact tool a drain person would try first, and renting one for a few hours will cost less than a plumber coming out to do it.

If that doesn't work, you'll soon be spending more than what a plumber would charge buying stuff that you don't know how to use in order to maybe fix the problem.

A drain company will have several snakes, an inspection camera, a pressure-washer based drain clearing device, etc, and they can change techniques every time one of them fails. Going through the same process buying/renting stuff every step of the way will quickly get expensive.

One of the other problems with drain clearing is that there is some voodoo involved in reasoning about how the system is put together and where your snake is going. Plumbers have intuition about this based on the age of the building, materials used, construction details, etc, but you likely don't and it's not really easy to teach that knowledge. Sometimes the right way to get to a certain part of the system with a snake is not intuitive.

Good luck.