Your problems are likely more than you realize. By your description, the water level gets higher than the concrete slab. If this has been happening for years you've probably got a rotted sill, which is the where the wood sits on the concrete.
The wood will wick up water above the water line, inside the wall and you'll get rot and mold in and on the walls.
I'm going to guess that over the years, erosion from the hill has left the ground level behind the garage higher than the slab elevation, and you've probably got a negative grade, meaning the dirt from the hill slopes to the garage, not away from the garage. You could try to remove enough dirt from behind the garage to create a downward slope away from the garage to the bottom of the hill, making sure you've got several inches of the slab exposed. Might be a lot of dirt, and you might want to consider installing a low retaining wall along the hill to prevent future erosion.
Create a shallow ditch at the base of the hill, at the bottom of the grade coming away from the garage, creating a channel for the water, sloped to either or both sides of the garage. It sounds like enough water comes down the hill that you'll need some type of drainage. If you can't channel the water around the garage, you may have to install a drainage pipe, given that you've got a downhill destination for the pipe.
Don't recall the recommended slopes for a positive grade against a building, something like 1 inch every 4 feet, or for a drain pipe, something like 1/8th inch every 4 feet, but these should be easy to find.
Also, if the roof of the garage contributes any water at all, install gutters. If your garage and your neighbors garage, or your house are close to gether, and the roofs slope towards each other, you could be getting a lot of the water simply from roof runoff. If there are gutters, make sure the downspouts are at the front of the garage.
I wouldn't suggest any type of outside sump pump unless you absolutely cannot find a way to let gravity move the water.
Otherwise, when you sell the property, advertise an indoor water feature. :D